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

Visit K11n4 at Knotilus!

Knot K11n4.
A graph, knot K11n4.
A part of a knot and a part of a graph.

Knot presentations

Planar diagram presentation X4251 X8394 X10,6,11,5 X14,8,15,7 X2,9,3,10 X18,12,19,11 X6,14,7,13 X15,20,16,21 X12,18,13,17 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 K11n4 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n4/ThurstonBennequinNumber
Hyperbolic Volume 12.5531
A-Polynomial See Data:K11n4/A-polynomial

[edit Notes for K11n4's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n4's four dimensional invariants]

Polynomial invariants

Alexander polynomial
Conway polynomial
2nd Alexander ideal (db, data sources)
Determinant and Signature { 49, 0 }
Jones polynomial
HOMFLY-PT polynomial (db, data sources)
Kauffman polynomial (db, data sources)
The A2 invariant Data:K11n4/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n4/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {9_27, K11n21, K11n172,}

Same Jones Polynomial (up to mirroring, ): {9_41, K11n21,}

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

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 are shown, along with their alternating sums (fixed , alternation over ). The squares with yellow highlighting are those on the "critical diagonals", where or , where 0 is the signature of K11n4. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
13         11
11        2 -2
9       31 2
7      42  -2
5     43   1
3    44    0
1   44     0
-1  25      3
-3 13       -2
-5 2        2
-71         -1
Integral Khovanov Homology

(db, data source)


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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