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HOMFLY polynomial
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<h2 id="definition">Definition</h2> <p>The polynomial is defined using <a href="/facts/Skein_relation/7LbCnnLp">skein relations</a>: </p> P ( u n k n o t ) = 1 , {\displaystyle P(\mathrm {unknot} )=1,\,} ℓ P ( L + ) + ℓ − 1 P ( L − ) + m P ( L 0 ) = 0 , {\displaystyle \ell P(L_{+})+\ell ^{-1}P(L_{-})+mP(L_{0})=0,\,} <p> where L + , L − , L 0 {\displaystyle L_{+},L_{-},L_{0}} are links formed by crossing and smoothing changes on a local region of a link diagram, as indicated in the figure. </p> <p>The HOMFLY polynomial of a link <i>L</i> that is a split union of two links L 1 {\displaystyle L_{1}} and L 2 {\displaystyle L_{2}} is given by </p> P ( L ) = − ( ℓ + ℓ − 1 ) m P ( L 1 ) P ( L 2 ) . {\displaystyle P(L)={\frac {-(\ell +\ell ^{-1})}{m}}P(L_{1})P(L_{2}).} <p>See the page on <a href="/facts/Skein_relation/7LbCnnLp">skein relation</a> for an example of a computation using such relations. </p> <h2 id="other-homfly-skein-relations">Other HOMFLY skein relations</h2> <p>This polynomial can be obtained also using other skein relations: </p> α P ( L + ) − α − 1 P ( L − ) = z P ( L 0 ) , {\displaystyle \alpha P(L_{+})-\alpha ^{-1}P(L_{-})=zP(L_{0}),\,} x P ( L + ) + y P ( L − ) + z P ( L 0 ) = 0 , {\displaystyle xP(L_{+})+yP(L_{-})+zP(L_{0})=0,\,} <h2 id="main-properties">Main properties</h2> P ( L 1 # L 2 ) = P ( L 1 ) P ( L 2 ) , {\displaystyle P(L_{1}\#L_{2})=P(L_{1})P(L_{2}),\,} , where # denotes the <a href="/facts/Knot_sum/TycjIkQe">knot sum</a>; thus the HOMFLY polynomial of a <a href="/facts/Composite_knot/Chg8aTHe">composite knot</a> is the product of the HOMFLY polynomials of its components. P K ( ℓ , m ) = P Mirror Image ( K ) ( ℓ − 1 , m ) {\displaystyle P_{K}(\ell ,m)=P_{{\text{Mirror Image}}(K)}(\ell ^{-1},m)} , so the HOMFLY polynomial can often be used to distinguish between two knots of different <a href="/facts/Chirality/hjx0eGrB">chirality</a>. However there exist chiral pairs of knots that have the same HOMFLY polynomial, e.g. knots 942 and 1071 together with their respective mirror images.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> <p>The Jones polynomial, <i>V</i>(<i>t</i>), and the Alexander polynomial, Δ ( t ) {\displaystyle \Delta (t)\,} can be computed in terms of the HOMFLY polynomial (the version in α {\displaystyle \alpha } and z {\displaystyle z} variables) as follows: </p> V ( t ) = P ( α = t − 1 , z = t 1 / 2 − t − 1 / 2 ) , {\displaystyle V(t)=P(\alpha =t^{-1},z=t^{1/2}-t^{-1/2}),\,} Δ ( t ) = P ( α = 1 , z = t 1 / 2 − t − 1 / 2 ) , {\displaystyle \Delta (t)=P(\alpha =1,z=t^{1/2}-t^{-1/2}),\,} <h2 id="further-reading">Further reading</h2> <ul><li><a href="/facts/Louis_Kauffman/lep7X1gB">Kauffman, L.H.</a>, "Formal knot theory", Princeton University Press, 1983.</li> <li><a href="/facts/W._B._R._Lickorish/s14Vg8FX">Lickorish, W.B.R.</a> "An Introduction to Knot Theory". Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-98254-X.</li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="https://www.encyclopediaofmath.org/index.php?title=Jones-Conway_polynomial">"Jones-Conway polynomial"</a>, <i><a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a></i>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a>, 2001 [1994]</li> <li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/HOMFLYPolynomial.html">"HOMFLY Polynomial"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li> <li>"<a href="https://katlas.org/wiki/The_HOMFLY-PT_Polynomial">The HOMFLY-PT Polynomial</a>", <i><a href="/facts/The_Knot_Atlas/LjAt3TEN">The Knot Atlas</a></i>.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Freyd, P.; Yetter, D.; Hoste, J.; Lickorish, W.B.R.; Millett, K.; Ocneanu, A. (1985). "A New Polynomial Invariant of Knots and Links". Bulletin of the American Mathematical Society. 12 (2): 239–246. doi:10.1090/S0273-0979-1985-15361-3. <a href="https://doi.org/10.1090%2FS0273-0979-1985-15361-3" target="_blank">https://doi.org/10.1090%2FS0273-0979-1985-15361-3</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Józef H. Przytycki; .Paweł Traczyk (1987). "Invariants of Links of Conway Type". Kobe J. Math. 4: 115–139. arXiv:1610.06679. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Ramadevi, P.; Govindarajan, T.R.; Kaul, R.K. (1994). "Chirality of Knots 942 and 1071 and Chern-Simons Theory". Modern Physics Letters A. 09 (34): 3205–3217. arXiv:hep-th/9401095. Bibcode:1994MPLA....9.3205R. doi:10.1142/S0217732394003026. S2CID 119143024. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> </ol>