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(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n134 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X5,19,6,18 X7,22,8,1 X9,14,10,15 X2,11,3,12 X13,20,14,21 X15,8,16,9 X17,7,18,6 X19,12,20,13 X21,16,22,17
Gauss code 1, -6, 2, -1, -3, 9, -4, 8, -5, -2, 6, 10, -7, 5, -8, 11, -9, 3, -10, 7, -11, 4
Dowker-Thistlethwaite code 4 10 -18 -22 -14 2 -20 -8 -6 -12 -16
A Braid Representative
A Morse Link Presentation K11n134 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number 1
3-genus 2
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n134/ThurstonBennequinNumber
Hyperbolic Volume 13.0489
A-Polynomial See Data:K11n134/A-polynomial

[edit Notes for K11n134's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 2
Rasmussen s-Invariant 2

[edit Notes for K11n134's four dimensional invariants]

Polynomial invariants

Alexander polynomial -3 t^2+12 t-17+12 t^{-1} -3 t^{-2}
Conway polynomial 1-3 z^4
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 47, -2 }
Jones polynomial 3 q^{-1} -5 q^{-2} +7 q^{-3} -8 q^{-4} +8 q^{-5} -7 q^{-6} +5 q^{-7} -3 q^{-8} + q^{-9}
HOMFLY-PT polynomial (db, data sources) z^2 a^8-z^4 a^6+a^6-2 z^4 a^4-4 z^2 a^4-3 a^4+3 z^2 a^2+3 a^2
Kauffman polynomial (db, data sources) z^6 a^{10}-3 z^4 a^{10}+2 z^2 a^{10}+3 z^7 a^9-10 z^5 a^9+8 z^3 a^9-z a^9+3 z^8 a^8-8 z^6 a^8+3 z^4 a^8+z^9 a^7+2 z^7 a^7-11 z^5 a^7+6 z^3 a^7-z a^7+4 z^8 a^6-9 z^6 a^6+4 z^4 a^6-a^6+z^9 a^5-z^7 a^5+2 z^5 a^5-3 z^3 a^5+z a^5+z^8 a^4-2 z^4 a^4+7 z^2 a^4-3 a^4+3 z^5 a^3-z^3 a^3+z a^3+5 z^2 a^2-3 a^2
The A2 invariant Data:K11n134/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n134/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {9_25,}

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

Vassiliev invariants

V2 and V3: (0, 1)
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
0 8 0 0 24 0 -\frac{304}{3} -\frac{256}{3} -56 0 32 0 0 432 152 280 16 0

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=-2 is the signature of K11n134. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
-1        33
-3       31-2
-5      42 2
-7     43  -1
-9    44   0
-11   34    1
-13  24     -2
-15 13      2
-17 2       -2
-191        1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-3 i=-1
r=-8 {\mathbb Z}
r=-7 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=0 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}

Computer Talk

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

Modifying This Page

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