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

Visit K11a188 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X12,3,13,4 X14,6,15,5 X16,8,17,7 X20,9,21,10 X18,11,19,12 X2,13,3,14 X6,16,7,15 X22,18,1,17 X10,19,11,20 X8,21,9,22
Gauss code 1, -7, 2, -1, 3, -8, 4, -11, 5, -10, 6, -2, 7, -3, 8, -4, 9, -6, 10, -5, 11, -9
Dowker-Thistlethwaite code 4 12 14 16 20 18 2 6 22 10 8
A Braid Representative
A Morse Link Presentation K11a188 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 2
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a188/ThurstonBennequinNumber
Hyperbolic Volume 10.6589
A-Polynomial See Data:K11a188/A-polynomial

[edit Notes for K11a188'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 K11a188's four dimensional invariants]

Polynomial invariants

Alexander polynomial 2 t^3-9 t^2+15 t-15+15 t^{-1} -9 t^{-2} +2 t^{-3}
Conway polynomial 2 z^6+3 z^4-3 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 67, 2 }
Jones polynomial -q^6+3 q^5-5 q^4+8 q^3-9 q^2+10 q-10+8 q^{-1} -6 q^{-2} +4 q^{-3} -2 q^{-4} + q^{-5}
HOMFLY-PT polynomial (db, data sources) z^6 a^{-2} +z^6-2 a^2 z^4+3 z^4 a^{-2} -z^4 a^{-4} +3 z^4+a^4 z^2-6 a^2 z^2+2 z^2 a^{-2} -2 z^2 a^{-4} +2 z^2+2 a^4-3 a^2+ a^{-2} +1
Kauffman polynomial (db, data sources) a^2 z^{10}+z^{10}+2 a^3 z^9+5 a z^9+3 z^9 a^{-1} +a^4 z^8-a^2 z^8+5 z^8 a^{-2} +3 z^8-11 a^3 z^7-21 a z^7-3 z^7 a^{-1} +7 z^7 a^{-3} -6 a^4 z^6-13 a^2 z^6-7 z^6 a^{-2} +7 z^6 a^{-4} -21 z^6+19 a^3 z^5+24 a z^5-13 z^5 a^{-1} -13 z^5 a^{-3} +5 z^5 a^{-5} +12 a^4 z^4+30 a^2 z^4-8 z^4 a^{-2} -10 z^4 a^{-4} +3 z^4 a^{-6} +23 z^4-11 a^3 z^3-6 a z^3+15 z^3 a^{-1} +6 z^3 a^{-3} -3 z^3 a^{-5} +z^3 a^{-7} -9 a^4 z^2-19 a^2 z^2+8 z^2 a^{-2} +4 z^2 a^{-4} -z^2 a^{-6} -7 z^2+2 a^3 z-a z-5 z a^{-1} -2 z a^{-3} +2 a^4+3 a^2- a^{-2} +1
The A2 invariant Data:K11a188/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a188/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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

Vassiliev invariants

V2 and V3: (-3, 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
-12 16 72 66 14 -192 -\frac{992}{3} -\frac{416}{3} -16 -288 128 -792 -168 -\frac{3631}{10} -\frac{4694}{15} \frac{2578}{15} -\frac{17}{6} \frac{529}{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=2 is the signature of K11a188. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
13           1-1
11          2 2
9         31 -2
7        52  3
5       43   -1
3      65    1
1     55     0
-1    35      -2
-3   35       2
-5  13        -2
-7 13         2
-9 1          -1
-111           1
Integral Khovanov Homology

(db, data source)

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