# T(11,3)

Jump to: navigation, search

## Contents

 See other torus knots Visit T(11,3)'s page at Knotilus! Visit T(11,3)'s page at the original Knot Atlas! Edit T(11,3) Quick Notes

### Knot presentations

 Planar diagram presentation X7,37,8,36 X22,38,23,37 X23,9,24,8 X38,10,39,9 X39,25,40,24 X10,26,11,25 X11,41,12,40 X26,42,27,41 X27,13,28,12 X42,14,43,13 X43,29,44,28 X14,30,15,29 X15,1,16,44 X30,2,31,1 X31,17,32,16 X2,18,3,17 X3,33,4,32 X18,34,19,33 X19,5,20,4 X34,6,35,5 X35,21,36,20 X6,22,7,21 Gauss code 14, -16, -17, 19, 20, -22, -1, 3, 4, -6, -7, 9, 10, -12, -13, 15, 16, -18, -19, 21, 22, -2, -3, 5, 6, -8, -9, 11, 12, -14, -15, 17, 18, -20, -21, 1, 2, -4, -5, 7, 8, -10, -11, 13 Dowker-Thistlethwaite code 30 -32 34 -36 38 -40 42 -44 2 -4 6 -8 10 -12 14 -16 18 -20 22 -24 26 -28

### Polynomial invariants

 Alexander polynomial t10−t9 + t7−t6 + t4−t3 + t−1 + t−1−t−3 + t−4−t−6 + t−7−t−9 + t−10 Conway polynomial z20 + 19z18 + 152z16 + 666z14 + 1742z12 + 2782z10 + 2665z8 + 1443z6 + 390z4 + 40z2 + 1 2nd Alexander ideal (db, data sources) {1} Determinant and Signature { 1, 16 } Jones polynomial −q22 + q12 + q10 HOMFLY-PT polynomial (db, data sources) z20a−20 + 20z18a−20−z18a−22 + 171z16a−20−19z16a−22 + 817z14a−20−152z14a−22 + z14a−24 + 2394z12a−20−666z12a−22 + 14z12a−24 + 4446z10a−20−1742z10a−22 + 78z10a−24 + 5226z8a−20−2782z8a−22 + 221z8a−24 + 3770z6a−20−2665z6a−22 + 338z6a−24 + 1560z4a−20−1443z4a−22 + 273z4a−24 + 325z2a−20−390z2a−22 + 105z2a−24 + 26a−20−40a−22 + 15a−24 Kauffman polynomial (db, data sources) Data:T(11,3)/Kauffman Polynomial The A2 invariant Data:T(11,3)/QuantumInvariant/A2/1,0 The G2 invariant Data:T(11,3)/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: (40, 220)
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(11,3)/V 2,1 Data:T(11,3)/V 3,1 Data:T(11,3)/V 4,1 Data:T(11,3)/V 4,2 Data:T(11,3)/V 4,3 Data:T(11,3)/V 5,1 Data:T(11,3)/V 5,2 Data:T(11,3)/V 5,3 Data:T(11,3)/V 5,4 Data:T(11,3)/V 6,1 Data:T(11,3)/V 6,2 Data:T(11,3)/V 6,3 Data:T(11,3)/V 6,4 Data:T(11,3)/V 6,5 Data:T(11,3)/V 6,6 Data:T(11,3)/V 6,7 Data:T(11,3)/V 6,8 Data:T(11,3)/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 = 16 is the signature of T(11,3). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
0123456789101112131415χ
45               1-1
43             1  -1
41             11 0
39           11   0
37         1  1   0
35         11     0
33       11       0
31     1  1       0
29     11         0
27   11           0
25    1           1
23  1             1
211               1
191               1
Integral Khovanov Homology
 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ i = 13 i = 15 i = 17 i = 19 i = 21 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 ${\mathbb Z}$ r = 7 ${\mathbb Z}_2$ ${\mathbb Z}$ r = 8 ${\mathbb Z}$ ${\mathbb Z}$ r = 9 ${\mathbb Z}$ ${\mathbb Z}$ r = 10 ${\mathbb Z}$ r = 11 ${\mathbb Z}_2$ ${\mathbb Z}$ r = 12 ${\mathbb Z}$ ${\mathbb Z}$ r = 13 ${\mathbb Z}$ ${\mathbb Z}$ r = 14 ${\mathbb Z}$ r = 15 ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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

###  Modifying This Page

Read me first: Modifying Knot Pages

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

See/edit the Torus Knot_Splice_Base (expert).

Back to the top.