# T(8,5)

## Contents

 See other torus knots Visit T(8,5)'s page at Knotilus! Visit T(8,5)'s page at the original Knot Atlas! Edit T(8,5) Quick Notes

### Knot presentations

 Planar diagram presentation X54,16,55,15 X29,17,30,16 X4,18,5,17 X43,19,44,18 X30,56,31,55 X5,57,6,56 X44,58,45,57 X19,59,20,58 X6,32,7,31 X45,33,46,32 X20,34,21,33 X59,35,60,34 X46,8,47,7 X21,9,22,8 X60,10,61,9 X35,11,36,10 X22,48,23,47 X61,49,62,48 X36,50,37,49 X11,51,12,50 X62,24,63,23 X37,25,38,24 X12,26,13,25 X51,27,52,26 X38,64,39,63 X13,1,14,64 X52,2,53,1 X27,3,28,2 X14,40,15,39 X53,41,54,40 X28,42,29,41 X3,43,4,42 Gauss code 27, 28, -32, -3, -6, -9, 13, 14, 15, 16, -20, -23, -26, -29, 1, 2, 3, 4, -8, -11, -14, -17, 21, 22, 23, 24, -28, -31, -2, -5, 9, 10, 11, 12, -16, -19, -22, -25, 29, 30, 31, 32, -4, -7, -10, -13, 17, 18, 19, 20, -24, -27, -30, -1, 5, 6, 7, 8, -12, -15, -18, -21, 25, 26 Dowker-Thistlethwaite code 52 -42 -56 46 60 -50 -64 54 4 -58 -8 62 12 -2 -16 6 20 -10 -24 14 28 -18 -32 22 36 -26 -40 30 44 -34 -48 38

### Polynomial invariants

 Alexander polynomial t14−t13 + t9−t8 + t6−t5 + t4−t3 + t−1 + t−1−t−3 + t−4−t−5 + t−6−t−8 + t−9−t−13 + t−14 Conway polynomial z28 + 27z26 + 324z24 + 2277z22 + 10395z20 + 32320z18 + 69785z16 + 104771z14 + 107849z12 + 73876z10 + 32055z8 + 8169z6 + 1092z4 + 63z2 + 1 2nd Alexander ideal (db, data sources) {1} Determinant and Signature { 5, 20 } Jones polynomial −q25−q23 + q18 + q16 + q14 HOMFLY-PT polynomial (db, data sources) Data:T(8,5)/HOMFLYPT Polynomial Kauffman polynomial (db, data sources) Data:T(8,5)/Kauffman Polynomial The A2 invariant Data:T(8,5)/QuantumInvariant/A2/1,0 The G2 invariant Data:T(8,5)/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: (63, 420)
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(8,5)/V 2,1 Data:T(8,5)/V 3,1 Data:T(8,5)/V 4,1 Data:T(8,5)/V 4,2 Data:T(8,5)/V 4,3 Data:T(8,5)/V 5,1 Data:T(8,5)/V 5,2 Data:T(8,5)/V 5,3 Data:T(8,5)/V 5,4 Data:T(8,5)/V 6,1 Data:T(8,5)/V 6,2 Data:T(8,5)/V 6,3 Data:T(8,5)/V 6,4 Data:T(8,5)/V 6,5 Data:T(8,5)/V 6,6 Data:T(8,5)/V 6,7 Data:T(8,5)/V 6,8 Data:T(8,5)/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 trqj are shown, along with their alternating sums χ (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 = 20 is the signature of T(8,5). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
012345678910111213141516171819χ
57                  110
55                11  0
53                121 0
51              131   -1
49            12 11   -1
47             32     -1
45           32       -1
43         2  2       0
41       1 12         0
39     1 12           0
37     11 1           1
35   11 1             1
33    1               1
31  1                 1
291                   1
271                   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 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 ${\mathbb Z}^{2}$ ${\mathbb Z}_2$ ${\mathbb Z}_2$ r = 11 ${\mathbb Z}_2^{2}\oplus{\mathbb Z}_5$ ${\mathbb Z}^{3}$ r = 12 ${\mathbb Z}^{2}$ ${\mathbb Z}^{2}$ ${\mathbb Z}_2\oplus{\mathbb Z}_5$ ${\mathbb Z}$ r = 13 ${\mathbb Z}_2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ r = 14 ${\mathbb Z}^{2}$ ${\mathbb Z}_2$ ${\mathbb Z}$ r = 15 ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ r = 16 ${\mathbb Z}$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = 17 ${\mathbb Z}^{2}$ ${\mathbb Z}$ r = 18 ${\mathbb Z}$ ${\mathbb Z}_2$ ${\mathbb Z}$ r = 19 ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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

Read me first: Modifying Knot Pages

See/edit the Torus Knot Page master template (intermediate).

See/edit the Torus Knot_Splice_Base (expert).

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