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

Visit K11n130 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X5,18,6,19 X7,14,8,15 X9,17,10,16 X2,11,3,12 X13,21,14,20 X15,22,16,1 X17,13,18,12 X19,6,20,7 X21,9,22,8
Gauss code 1, -6, 2, -1, -3, 10, -4, 11, -5, -2, 6, 9, -7, 4, -8, 5, -9, 3, -10, 7, -11, 8
Dowker-Thistlethwaite code 4 10 -18 -14 -16 2 -20 -22 -12 -6 -8
A Braid Representative
A Morse Link Presentation K11n130 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number 1
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n130/ThurstonBennequinNumber
Hyperbolic Volume 13.3479
A-Polynomial See Data:K11n130/A-polynomial

[edit Notes for K11n130's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant 0

[edit Notes for K11n130's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^3+5 t^2-12 t+17-12 t^{-1} +5 t^{-2} - t^{-3}
Conway polynomial -z^6-z^4-z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 53, 0 }
Jones polynomial -q^3+4 q^2-6 q+8-9 q^{-1} +9 q^{-2} -7 q^{-3} +5 q^{-4} -3 q^{-5} + q^{-6}
HOMFLY-PT polynomial (db, data sources) -a^2 z^6+a^4 z^4-4 a^2 z^4+2 z^4+2 a^4 z^2-6 a^2 z^2-z^2 a^{-2} +4 z^2+a^4-2 a^2+2
Kauffman polynomial (db, data sources) 2 a^3 z^9+2 a z^9+4 a^4 z^8+7 a^2 z^8+3 z^8+3 a^5 z^7-3 a^3 z^7-5 a z^7+z^7 a^{-1} +a^6 z^6-14 a^4 z^6-25 a^2 z^6-10 z^6-10 a^5 z^5-5 a^3 z^5+7 a z^5+2 z^5 a^{-1} -3 a^6 z^4+13 a^4 z^4+31 a^2 z^4+4 z^4 a^{-2} +19 z^4+7 a^5 z^3+6 a^3 z^3-4 a z^3-2 z^3 a^{-1} +z^3 a^{-3} +a^6 z^2-5 a^4 z^2-15 a^2 z^2-3 z^2 a^{-2} -12 z^2-a^5 z-a^3 z+a^4+2 a^2+2
The A2 invariant q^{18}-q^{16}+q^{14}-2 q^{10}+2 q^8-q^6+q^4-1+2 q^{-2} - q^{-4} +2 q^{-6} + q^{-8} - q^{-10}
The G2 invariant Data:K11n130/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {9_30,}

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

Vassiliev invariants

V2 and V3: (-1, 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
-4 8 8 \frac{34}{3} \frac{38}{3} -32 -\frac{208}{3} -\frac{160}{3} 8 -\frac{32}{3} 32 -\frac{136}{3} -\frac{152}{3} \frac{2129}{30} \frac{1142}{15} -\frac{4742}{45} \frac{1039}{18} -\frac{751}{30}

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=0 is the signature of K11n130. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
7         1-1
5        3 3
3       31 -2
1      53  2
-1     54   -1
-3    44    0
-5   35     2
-7  24      -2
-9 13       2
-11 2        -2
-131         1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=0 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{5}
r=1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=3 {\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 Hoste-Thistlethwaite Knot Page master template (intermediate).

See/edit the Hoste-Thistlethwaite_Splice_Base (expert).

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