T(25,2)

Knot presentations

 Planar diagram presentation X23,49,24,48 X49,25,50,24 X25,1,26,50 X1,27,2,26 X27,3,28,2 X3,29,4,28 X29,5,30,4 X5,31,6,30 X31,7,32,6 X7,33,8,32 X33,9,34,8 X9,35,10,34 X35,11,36,10 X11,37,12,36 X37,13,38,12 X13,39,14,38 X39,15,40,14 X15,41,16,40 X41,17,42,16 X17,43,18,42 X43,19,44,18 X19,45,20,44 X45,21,46,20 X21,47,22,46 X47,23,48,22 Gauss code -4, 5, -6, 7, -8, 9, -10, 11, -12, 13, -14, 15, -16, 17, -18, 19, -20, 21, -22, 23, -24, 25, -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 12, -13, 14, -15, 16, -17, 18, -19, 20, -21, 22, -23, 24, -25, 1, -2, 3 Dowker-Thistlethwaite code 26 28 30 32 34 36 38 40 42 44 46 48 50 2 4 6 8 10 12 14 16 18 20 22 24

Polynomial invariants

 Alexander polynomial $t^{12}-t^{11}+t^{10}-t^9+t^8-t^7+t^6-t^5+t^4-t^3+t^2-t+1- t^{-1} + t^{-2} - t^{-3} + t^{-4} - t^{-5} + t^{-6} - t^{-7} + t^{-8} - t^{-9} + t^{-10} - t^{-11} + t^{-12}$ Conway polynomial $z^{24}+23 z^{22}+231 z^{20}+1330 z^{18}+4845 z^{16}+11628 z^{14}+18564 z^{12}+19448 z^{10}+12870 z^8+5005 z^6+1001 z^4+78 z^2+1$ 2nd Alexander ideal (db, data sources) $\{1\}$ Determinant and Signature { 25, 24 } Jones polynomial $-q^{37}+q^{36}-q^{35}+q^{34}-q^{33}+q^{32}-q^{31}+q^{30}-q^{29}+q^{28}-q^{27}+q^{26}-q^{25}+q^{24}-q^{23}+q^{22}-q^{21}+q^{20}-q^{19}+q^{18}-q^{17}+q^{16}-q^{15}+q^{14}+q^{12}$ HOMFLY-PT polynomial (db, data sources) $z^{24}a^{-24}-24z^{22}a^{-24}-z^{22}a^{-26}+253z^{20}a^{-24}+22z^{20}a^{-26}-1540z^{18}a^{-24}-210z^{18}a^{-26}+5985z^{16}a^{-24}+1140z^{16}a^{-26}-15504z^{14}a^{-24}-3876z^{14}a^{-26}+27132z^{12}a^{-24}+8568z^{12}a^{-26}-31824z^{10}a^{-24}-12376z^{10}a^{-26}+24310z^8a^{-24}+11440z^8a^{-26}-11440z^6a^{-24}-6435z^6a^{-26}+3003z^4a^{-24}+2002z^4a^{-26}-364z^2a^{-24}-286z^2a^{-26}+13a^{-24}+12a^{-26}$ Kauffman polynomial (db, data sources) $z^{24}a^{-24}+z^{24}a^{-26}+z^{23}a^{-25}+z^{23}a^{-27}-24z^{22}a^{-24}-23z^{22}a^{-26}+z^{22}a^{-28}-22z^{21}a^{-25}-21z^{21}a^{-27}+z^{21}a^{-29}+253z^{20}a^{-24}+232z^{20}a^{-26}-20z^{20}a^{-28}+z^{20}a^{-30}+210z^{19}a^{-25}+190z^{19}a^{-27}-19z^{19}a^{-29}+z^{19}a^{-31}-1540z^{18}a^{-24}-1350z^{18}a^{-26}+171z^{18}a^{-28}-18z^{18}a^{-30}+z^{18}a^{-32}-1140z^{17}a^{-25}-969z^{17}a^{-27}+153z^{17}a^{-29}-17z^{17}a^{-31}+z^{17}a^{-33}+5985z^{16}a^{-24}+5016z^{16}a^{-26}-816z^{16}a^{-28}+136z^{16}a^{-30}-16z^{16}a^{-32}+z^{16}a^{-34}+3876z^{15}a^{-25}+3060z^{15}a^{-27}-680z^{15}a^{-29}+120z^{15}a^{-31}-15z^{15}a^{-33}+z^{15}a^{-35}-15504z^{14}a^{-24}-12444z^{14}a^{-26}+2380z^{14}a^{-28}-560z^{14}a^{-30}+105z^{14}a^{-32}-14z^{14}a^{-34}+z^{14}a^{-36}-8568z^{13}a^{-25}-6188z^{13}a^{-27}+1820z^{13}a^{-29}-455z^{13}a^{-31}+91z^{13}a^{-33}-13z^{13}a^{-35}+z^{13}a^{-37}+27132z^{12}a^{-24}+20944z^{12}a^{-26}-4368z^{12}a^{-28}+1365z^{12}a^{-30}-364z^{12}a^{-32}+78z^{12}a^{-34}-12z^{12}a^{-36}+z^{12}a^{-38}+12376z^{11}a^{-25}+8008z^{11}a^{-27}-3003z^{11}a^{-29}+1001z^{11}a^{-31}-286z^{11}a^{-33}+66z^{11}a^{-35}-11z^{11}a^{-37}+z^{11}a^{-39}-31824z^{10}a^{-24}-23816z^{10}a^{-26}+5005z^{10}a^{-28}-2002z^{10}a^{-30}+715z^{10}a^{-32}-220z^{10}a^{-34}+55z^{10}a^{-36}-10z^{10}a^{-38}+z^{10}a^{-40}-11440z^9a^{-25}-6435z^9a^{-27}+3003z^9a^{-29}-1287z^9a^{-31}+495z^9a^{-33}-165z^9a^{-35}+45z^9a^{-37}-9z^9a^{-39}+z^9a^{-41}+24310z^8a^{-24}+17875z^8a^{-26}-3432z^8a^{-28}+1716z^8a^{-30}-792z^8a^{-32}+330z^8a^{-34}-120z^8a^{-36}+36z^8a^{-38}-8z^8a^{-40}+z^8a^{-42}+6435z^7a^{-25}+3003z^7a^{-27}-1716z^7a^{-29}+924z^7a^{-31}-462z^7a^{-33}+210z^7a^{-35}-84z^7a^{-37}+28z^7a^{-39}-7z^7a^{-41}+z^7a^{-43}-11440z^6a^{-24}-8437z^6a^{-26}+1287z^6a^{-28}-792z^6a^{-30}+462z^6a^{-32}-252z^6a^{-34}+126z^6a^{-36}-56z^6a^{-38}+21z^6a^{-40}-6z^6a^{-42}+z^6a^{-44}-2002z^5a^{-25}-715z^5a^{-27}+495z^5a^{-29}-330z^5a^{-31}+210z^5a^{-33}-126z^5a^{-35}+70z^5a^{-37}-35z^5a^{-39}+15z^5a^{-41}-5z^5a^{-43}+z^5a^{-45}+3003z^4a^{-24}+2288z^4a^{-26}-220z^4a^{-28}+165z^4a^{-30}-120z^4a^{-32}+84z^4a^{-34}-56z^4a^{-36}+35z^4a^{-38}-20z^4a^{-40}+10z^4a^{-42}-4z^4a^{-44}+z^4a^{-46}+286z^3a^{-25}+66z^3a^{-27}-55z^3a^{-29}+45z^3a^{-31}-36z^3a^{-33}+28z^3a^{-35}-21z^3a^{-37}+15z^3a^{-39}-10z^3a^{-41}+6z^3a^{-43}-3z^3a^{-45}+z^3a^{-47}-364z^2a^{-24}-298z^2a^{-26}+11z^2a^{-28}-10z^2a^{-30}+9z^2a^{-32}-8z^2a^{-34}+7z^2a^{-36}-6z^2a^{-38}+5z^2a^{-40}-4z^2a^{-42}+3z^2a^{-44}-2z^2a^{-46}+z^2a^{-48}-12za^{-25}-za^{-27}+za^{-29}-za^{-31}+za^{-33}-za^{-35}+za^{-37}-za^{-39}+za^{-41}-za^{-43}+za^{-45}-za^{-47}+za^{-49}+13a^{-24}+12a^{-26}$ The A2 invariant Data:T(25,2)/QuantumInvariant/A2/1,0 The G2 invariant Data:T(25,2)/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {}

Vassiliev invariants

 V2 and V3: (78, 650)
V2,1 through V6,9:
 V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 Data:T(25,2)/V 2,1 Data:T(25,2)/V 3,1 Data:T(25,2)/V 4,1 Data:T(25,2)/V 4,2 Data:T(25,2)/V 4,3 Data:T(25,2)/V 5,1 Data:T(25,2)/V 5,2 Data:T(25,2)/V 5,3 Data:T(25,2)/V 5,4 Data:T(25,2)/V 6,1 Data:T(25,2)/V 6,2 Data:T(25,2)/V 6,3 Data:T(25,2)/V 6,4 Data:T(25,2)/V 6,5 Data:T(25,2)/V 6,6 Data:T(25,2)/V 6,7 Data:T(25,2)/V 6,8 Data:T(25,2)/V 6,9

V2,1 through V6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V2 and V3.

Khovanov Homology

The coefficients of the monomials $t^rq^j$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). The squares with yellow highlighting are those on the "critical diagonals", where $j-2r=s+1$ or $j-2r=s-1$, where $s=$24 is the signature of T(25,2). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
012345678910111213141516171819202122232425χ
75                         1-1
73                          0
71                       11 0
69                          0
67                     11   0
65                          0
63                   11     0
61                          0
59                 11       0
57                          0
55               11         0
53                          0
51             11           0
49                          0
47           11             0
45                          0
43         11               0
41                          0
39       11                 0
37                          0
35     11                   0
33                          0
31   11                     0
29                          0
27  1                       1
251                         1
231                         1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=23$ $i=25$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$ $r=1$ $r=2$ ${\mathbb Z}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=4$ ${\mathbb Z}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=6$ ${\mathbb Z}$ $r=7$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=8$ ${\mathbb Z}$ $r=9$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=10$ ${\mathbb Z}$ $r=11$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=12$ ${\mathbb Z}$ $r=13$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=14$ ${\mathbb Z}$ $r=15$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=16$ ${\mathbb Z}$ $r=17$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=18$ ${\mathbb Z}$ $r=19$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=20$ ${\mathbb Z}$ $r=21$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=22$ ${\mathbb Z}$ $r=23$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=24$ ${\mathbb Z}$ $r=25$ ${\mathbb Z}_2$ ${\mathbb Z}$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory. See A Sample KnotTheory Session.