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

Visit K11a128 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X14,5,15,6 X20,8,21,7 X16,10,17,9 X2,11,3,12 X6,13,7,14 X12,16,13,15 X22,17,1,18 X8,20,9,19 X18,21,19,22
Gauss code 1, -6, 2, -1, 3, -7, 4, -10, 5, -2, 6, -8, 7, -3, 8, -5, 9, -11, 10, -4, 11, -9
Dowker-Thistlethwaite code 4 10 14 20 16 2 6 12 22 8 18
A Braid Representative
A Morse Link Presentation K11a128 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a128/ThurstonBennequinNumber
Hyperbolic Volume 15.6943
A-Polynomial See Data:K11a128/A-polynomial

[edit Notes for K11a128's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant 0

[edit Notes for K11a128's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^3+9 t^2-31 t+47-31 t^{-1} +9 t^{-2} - t^{-3}
Conway polynomial -z^6+3 z^4-4 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 129, 0 }
Jones polynomial -q^5+4 q^4-8 q^3+14 q^2-18 q+21-21 q^{-1} +17 q^{-2} -13 q^{-3} +8 q^{-4} -3 q^{-5} + q^{-6}
HOMFLY-PT polynomial (db, data sources) a^6-3 z^2 a^4-a^4+3 z^4 a^2+3 z^2 a^2+a^2-z^6-2 z^4-5 z^2-2+2 z^4 a^{-2} +2 z^2 a^{-2} +2 a^{-2} -z^2 a^{-4}
Kauffman polynomial (db, data sources) a^2 z^{10}+z^{10}+4 a^3 z^9+8 a z^9+4 z^9 a^{-1} +5 a^4 z^8+13 a^2 z^8+7 z^8 a^{-2} +15 z^8+3 a^5 z^7-a^3 z^7-2 a z^7+9 z^7 a^{-1} +7 z^7 a^{-3} +a^6 z^6-10 a^4 z^6-31 a^2 z^6-4 z^6 a^{-2} +4 z^6 a^{-4} -28 z^6-7 a^5 z^5-10 a^3 z^5-18 a z^5-26 z^5 a^{-1} -10 z^5 a^{-3} +z^5 a^{-5} -3 a^6 z^4+5 a^4 z^4+21 a^2 z^4-7 z^4 a^{-2} -6 z^4 a^{-4} +12 z^4+5 a^5 z^3+5 a^3 z^3+10 a z^3+16 z^3 a^{-1} +5 z^3 a^{-3} -z^3 a^{-5} +3 a^6 z^2-5 a^2 z^2+7 z^2 a^{-2} +3 z^2 a^{-4} +2 z^2-a^5 z+2 a^3 z+2 a z-2 z a^{-1} -z a^{-3} -a^6-a^4-a^2-2 a^{-2} -2
The A2 invariant Data:K11a128/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a128/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: (-4, 3)
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
-16 24 128 \frac{376}{3} \frac{128}{3} -384 -560 -128 -104 -\frac{2048}{3} 288 -\frac{6016}{3} -\frac{2048}{3} -\frac{15062}{15} \frac{368}{15} -\frac{35408}{45} \frac{1526}{9} -\frac{2822}{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=0 is the signature of K11a128. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
11           1-1
9          3 3
7         51 -4
5        93  6
3       95   -4
1      129    3
-1     1010     0
-3    711      -4
-5   610       4
-7  27        -5
-9 16         5
-11 2          -2
-131           1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-2 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-1 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=0 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{12}
r=1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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

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See/edit the Hoste-Thistlethwaite_Splice_Base (expert).

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