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

Visit K11n83 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X8394 X5,17,6,16 X7,12,8,13 X2,9,3,10 X11,18,12,19 X13,1,14,22 X15,21,16,20 X17,10,18,11 X19,7,20,6 X21,15,22,14
Gauss code 1, -5, 2, -1, -3, 10, -4, -2, 5, 9, -6, 4, -7, 11, -8, 3, -9, 6, -10, 8, -11, 7
Dowker-Thistlethwaite code 4 8 -16 -12 2 -18 -22 -20 -10 -6 -14
A Braid Representative
A Morse Link Presentation K11n83 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2\}
3-genus 2
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n83/ThurstonBennequinNumber
Hyperbolic Volume 12.0122
A-Polynomial See Data:K11n83/A-polynomial

[edit Notes for K11n83's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus [0,2]
Rasmussen s-Invariant 0

[edit Notes for K11n83's four dimensional invariants]

Polynomial invariants

Alexander polynomial 3 t^2-12 t+19-12 t^{-1} +3 t^{-2}
Conway polynomial 3 z^4+1
2nd Alexander ideal (db, data sources) \left\{2,t^2+t+1\right\}
Determinant and Signature { 49, 0 }
Jones polynomial q^4-3 q^3+5 q^2-7 q+9-8 q^{-1} +7 q^{-2} -5 q^{-3} +3 q^{-4} - q^{-5}
HOMFLY-PT polynomial (db, data sources) -z^2 a^4+z^4 a^2-a^2+2 z^4+4 z^2+4-3 z^2 a^{-2} -3 a^{-2} + a^{-4}
Kauffman polynomial (db, data sources) a z^9+z^9 a^{-1} +3 a^2 z^8+z^8 a^{-2} +4 z^8+4 a^3 z^7+a z^7-3 z^7 a^{-1} +3 a^4 z^6-7 a^2 z^6-3 z^6 a^{-2} -13 z^6+a^5 z^5-10 a^3 z^5-5 a z^5+9 z^5 a^{-1} +3 z^5 a^{-3} -7 a^4 z^4+7 a^2 z^4+10 z^4 a^{-2} +z^4 a^{-4} +23 z^4-2 a^5 z^3+6 a^3 z^3+6 a z^3-7 z^3 a^{-1} -5 z^3 a^{-3} +2 a^4 z^2-4 a^2 z^2-11 z^2 a^{-2} -2 z^2 a^{-4} -15 z^2-a^3 z-a z+z a^{-1} +z a^{-3} +a^2+3 a^{-2} + a^{-4} +4
The A2 invariant Data:K11n83/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n83/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {9_41,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {9_27,}

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 -32 -24 0 -\frac{80}{3} \frac{160}{3} -72 0 32 0 0 32 \frac{232}{3} -24 -48 -32

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 K11n83. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
9         11
7        2 -2
5       31 2
3      42  -2
1     53   2
-1    45    1
-3   34     -1
-5  24      2
-7 13       -2
-9 2        2
-111         -1
Integral Khovanov Homology

(db, data source)

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