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

Visit K11n5 at Knotilus!

Knot K11n5.
A graph, knot K11n5.
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 X11,18,12,19 X6,14,7,13 X20,16,21,15 X17,12,18,13 X22,20,1,19 X16,22,17,21
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 K11n5 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number \{1,2\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n5/ThurstonBennequinNumber
Hyperbolic Volume 14.1156
A-Polynomial See Data:K11n5/A-polynomial

[edit Notes for K11n5's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant -2

[edit Notes for K11n5's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^3-7 t^2+17 t-21+17 t^{-1} -7 t^{-2} + t^{-3}
Conway polynomial z^6-z^4-2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 71, 2 }
Jones polynomial -q^8+4 q^7-7 q^6+10 q^5-12 q^4+12 q^3-11 q^2+8 q-4+2 q^{-1}
HOMFLY-PT polynomial (db, data sources) z^6 a^{-4} -3 z^4 a^{-2} +3 z^4 a^{-4} -z^4 a^{-6} -7 z^2 a^{-2} +4 z^2 a^{-4} -z^2 a^{-6} +2 z^2-4 a^{-2} +2 a^{-4} +3
Kauffman polynomial (db, data sources) z^9 a^{-3} +z^9 a^{-5} +2 z^8 a^{-2} +6 z^8 a^{-4} +4 z^8 a^{-6} +z^7 a^{-1} +4 z^7 a^{-3} +9 z^7 a^{-5} +6 z^7 a^{-7} -3 z^6 a^{-2} -8 z^6 a^{-4} -z^6 a^{-6} +4 z^6 a^{-8} +z^5 a^{-1} -9 z^5 a^{-3} -22 z^5 a^{-5} -11 z^5 a^{-7} +z^5 a^{-9} +10 z^4 a^{-2} +4 z^4 a^{-4} -10 z^4 a^{-6} -7 z^4 a^{-8} +3 z^4-z^3 a^{-1} +11 z^3 a^{-3} +17 z^3 a^{-5} +4 z^3 a^{-7} -z^3 a^{-9} -12 z^2 a^{-2} -2 z^2 a^{-4} +6 z^2 a^{-6} +2 z^2 a^{-8} -6 z^2-z a^{-1} -5 z a^{-3} -5 z a^{-5} -z a^{-7} +4 a^{-2} +2 a^{-4} +3
The A2 invariant Data:K11n5/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n5/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_41,}

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

Vassiliev invariants

V2 and V3: (-2, -2)
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
-8 -16 32 \frac{116}{3} \frac{76}{3} 128 \frac{608}{3} \frac{128}{3} 48 -\frac{256}{3} 128 -\frac{928}{3} -\frac{608}{3} -\frac{151}{15} \frac{308}{5} -\frac{10204}{45} \frac{535}{9} -\frac{1111}{15}

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 K11n5. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
17         1-1
15        3 3
13       41 -3
11      63  3
9     64   -2
7    66    0
5   56     1
3  36      -3
1 26       4
-1 2        -2
-32         2
Integral Khovanov Homology

(db, data source)

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