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

Visit K11n8 at Knotilus!

Knot K11n8.
A graph, knot K11n8.
A part of a knot and a part of a graph.

Knot presentations

Planar diagram presentation X4251 X8394 X10,6,11,5 X7,14,8,15 X2,9,3,10 X11,18,12,19 X13,6,14,7 X15,20,16,21 X17,12,18,13 X19,22,20,1 X21,16,22,17
Gauss code 1, -5, 2, -1, 3, 7, -4, -2, 5, -3, -6, 9, -7, 4, -8, 11, -9, 6, -10, 8, -11, 10
Dowker-Thistlethwaite code 4 8 10 -14 2 -18 -6 -20 -12 -22 -16
A Braid Representative
A Morse Link Presentation K11n8 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number 2
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n8/ThurstonBennequinNumber
Hyperbolic Volume 13.098
A-Polynomial See Data:K11n8/A-polynomial

[edit Notes for K11n8's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n8's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^3+6 t^2-12 t+15-12 t^{-1} +6 t^{-2} - t^{-3}
Conway polynomial -z^6+3 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 53, -4 }
Jones polynomial 1-3 q^{-1} +6 q^{-2} -7 q^{-3} +9 q^{-4} -9 q^{-5} +8 q^{-6} -6 q^{-7} +3 q^{-8} - q^{-9}
HOMFLY-PT polynomial (db, data sources) -z^2 a^8-a^8+2 z^4 a^6+4 z^2 a^6+a^6-z^6 a^4-3 z^4 a^4-2 z^2 a^4+z^4 a^2+2 z^2 a^2+a^2
Kauffman polynomial (db, data sources) z^3 a^{11}-z a^{11}+3 z^4 a^{10}-2 z^2 a^{10}+z^7 a^9+3 z^3 a^9-z a^9+2 z^8 a^8-4 z^6 a^8+6 z^4 a^8-a^8+z^9 a^7+2 z^7 a^7-9 z^5 a^7+10 z^3 a^7-3 z a^7+5 z^8 a^6-11 z^6 a^6+3 z^4 a^6+3 z^2 a^6-a^6+z^9 a^5+4 z^7 a^5-18 z^5 a^5+14 z^3 a^5-4 z a^5+3 z^8 a^4-6 z^6 a^4-3 z^4 a^4+4 z^2 a^4+3 z^7 a^3-9 z^5 a^3+6 z^3 a^3-z a^3+z^6 a^2-3 z^4 a^2+3 z^2 a^2-a^2
The A2 invariant Data:K11n8/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n8/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n59,}

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

Vassiliev invariants

V2 and V3: (3, -6)
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
12 -48 72 222 34 -576 -1120 -192 -144 288 1152 2664 408 \frac{58191}{10} \frac{3734}{15} \frac{31742}{15} \frac{209}{6} \frac{2671}{10}

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=-4 is the signature of K11n8. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
1         11
-1        2 -2
-3       41 3
-5      43  -1
-7     53   2
-9    44    0
-11   45     -1
-13  24      2
-15 14       -3
-17 2        2
-191         -1
Integral Khovanov Homology

(db, data source)

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