Notice. The Knot Atlas is now recovering from a major crash. Hopefully all functionality will return slowly over the next few days. --Drorbn (talk) 21:23, 4 July 2013 (EDT)

# T(7,6)

 See other torus knots Visit T(7,6)'s page at Knotilus! Visit T(7,6)'s page at the original Knot Atlas! Edit T(7,6) Quick Notes

### Knot presentations

 Planar diagram presentation X53,65,54,64 X42,66,43,65 X31,67,32,66 X20,68,21,67 X9,69,10,68 X43,55,44,54 X32,56,33,55 X21,57,22,56 X10,58,11,57 X69,59,70,58 X33,45,34,44 X22,46,23,45 X11,47,12,46 X70,48,1,47 X59,49,60,48 X23,35,24,34 X12,36,13,35 X1,37,2,36 X60,38,61,37 X49,39,50,38 X13,25,14,24 X2,26,3,25 X61,27,62,26 X50,28,51,27 X39,29,40,28 X3,15,4,14 X62,16,63,15 X51,17,52,16 X40,18,41,17 X29,19,30,18 X63,5,64,4 X52,6,53,5 X41,7,42,6 X30,8,31,7 X19,9,20,8 Gauss code -18, -22, -26, 31, 32, 33, 34, 35, -5, -9, -13, -17, -21, 26, 27, 28, 29, 30, -35, -4, -8, -12, -16, 21, 22, 23, 24, 25, -30, -34, -3, -7, -11, 16, 17, 18, 19, 20, -25, -29, -33, -2, -6, 11, 12, 13, 14, 15, -20, -24, -28, -32, -1, 6, 7, 8, 9, 10, -15, -19, -23, -27, -31, 1, 2, 3, 4, 5, -10, -14 Dowker-Thistlethwaite code 36 14 -52 -30 68 46 24 -62 -40 8 56 34 -2 -50 18 66 44 -12 -60 28 6 54 -22 -70 38 16 64 -32 -10 48 26 4 -42 -20 58

### Polynomial invariants

 Alexander polynomial $t^{15}-t^{14}+t^9-t^7+t^3-1+ t^{-3} - t^{-7} + t^{-9} - t^{-14} + t^{-15}$ Conway polynomial $z^{30}+29 z^{28}+377 z^{26}+2900 z^{24}+14674 z^{22}+51359 z^{20}+127282 z^{18}+224826 z^{16}+281144 z^{14}+244074 z^{12}+142208 z^{10}+52844 z^8+11649 z^6+1365 z^4+70 z^2+1$ 2nd Alexander ideal (db, data sources) $\{1\}$ Determinant and Signature { 7, 18 } Jones polynomial $-q^{26}-q^{24}-q^{22}+q^{21}+q^{19}+q^{17}+q^{15}$ HOMFLY-PT polynomial (db, data sources) Data:T(7,6)/HOMFLYPT Polynomial Kauffman polynomial (db, data sources) Data:T(7,6)/Kauffman Polynomial The A2 invariant Data:T(7,6)/QuantumInvariant/A2/1,0 The G2 invariant Data:T(7,6)/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: (70, 490)
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(7,6)/V 2,1 Data:T(7,6)/V 3,1 Data:T(7,6)/V 4,1 Data:T(7,6)/V 4,2 Data:T(7,6)/V 4,3 Data:T(7,6)/V 5,1 Data:T(7,6)/V 5,2 Data:T(7,6)/V 5,3 Data:T(7,6)/V 5,4 Data:T(7,6)/V 6,1 Data:T(7,6)/V 6,2 Data:T(7,6)/V 6,3 Data:T(7,6)/V 6,4 Data:T(7,6)/V 6,5 Data:T(7,6)/V 6,6 Data:T(7,6)/V 6,7 Data:T(7,6)/V 6,8 Data:T(7,6)/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=$18 is the signature of T(7,6). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
012345678910111213141516171819χ
57                1  10
55                11  0
53              12 11 -1
51            11 21   -1
49             31 1   -1
47           31 1     -1
45         2 12       -1
43       1 12         0
41     1 12 1         1
39     11 1           1
37   11 1             1
35    1               1
33  1                 1
311                   1
291                   1
Integral Khovanov Homology
 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=17$ $i=19$ $i=21$ $i=23$ $i=25$ $i=27$ $i=29$ $i=31$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$ $r=1$ $r=2$ ${\mathbb Z}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=4$ ${\mathbb Z}$ ${\mathbb Z}$ $r=5$ ${\mathbb Z}$ ${\mathbb Z}$ $r=6$ ${\mathbb Z}$ ${\mathbb Z}$ $r=7$ ${\mathbb Z}_2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=8$ ${\mathbb Z}$ ${\mathbb Z}^{2}$ $r=9$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=10$ ${\mathbb Z}$ ${\mathbb Z}^{2}$ ${\mathbb Z}_2$ ${\mathbb Z}_2$ $r=11$ ${\mathbb Z}_2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=12$ ${\mathbb Z}^{2}$ ${\mathbb Z}$ ${\mathbb Z}_2\oplus{\mathbb Z}_5$ ${\mathbb Z}$ $r=13$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=14$ ${\mathbb Z}$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}$ $r=15$ ${\mathbb Z}_2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=16$ ${\mathbb Z}$ ${\mathbb Z}$ ${\mathbb Z}_2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=17$ ${\mathbb Z}_2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=18$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}_4$ $r=19$ ${\mathbb Z}_2\oplus{\mathbb Z}_3$ ${\mathbb Z}$ $r=20$ ${\mathbb Z}_2$ ${\mathbb Z}_2\oplus{\mathbb Z}_3$ ${\mathbb Z}_3$ $r=21$ ${\mathbb Z}_2$ ${\mathbb Z}_2$

### Computer Talk

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