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

Visit K11n128 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X5,19,6,18 X7,15,8,14 X9,16,10,17 X2,11,3,12 X13,7,14,6 X15,20,16,21 X17,1,18,22 X19,12,20,13 X21,9,22,8
Gauss code 1, -6, 2, -1, -3, 7, -4, 11, -5, -2, 6, 10, -7, 4, -8, 5, -9, 3, -10, 8, -11, 9
Dowker-Thistlethwaite code 4 10 -18 -14 -16 2 -6 -20 -22 -12 -8
A Braid Representative
A Morse Link Presentation K11n128 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:K11n128/ThurstonBennequinNumber
Hyperbolic Volume 12.2619
A-Polynomial See Data:K11n128/A-polynomial

[edit Notes for K11n128'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 K11n128's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^3-5 t^2+10 t-11+10 t^{-1} -5 t^{-2} + t^{-3}
Conway polynomial z^6+z^4-z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 43, 2 }
Jones polynomial q^5-3 q^4+5 q^3-6 q^2+7 q-7+6 q^{-1} -4 q^{-2} +3 q^{-3} - q^{-4}
HOMFLY-PT polynomial (db, data sources) z^6-a^2 z^4-2 z^4 a^{-2} +4 z^4-2 a^2 z^2-5 z^2 a^{-2} +z^2 a^{-4} +5 z^2-2 a^{-2} + a^{-4} +2
Kauffman polynomial (db, data sources) 2 a z^9+2 z^9 a^{-1} +3 a^2 z^8+4 z^8 a^{-2} +7 z^8+a^3 z^7-6 a z^7-4 z^7 a^{-1} +3 z^7 a^{-3} -14 a^2 z^6-15 z^6 a^{-2} +z^6 a^{-4} -30 z^6-4 a^3 z^5-a z^5-5 z^5 a^{-1} -8 z^5 a^{-3} +18 a^2 z^4+19 z^4 a^{-2} +z^4 a^{-4} +36 z^4+4 a^3 z^3+8 a z^3+8 z^3 a^{-1} +7 z^3 a^{-3} +3 z^3 a^{-5} -6 a^2 z^2-12 z^2 a^{-2} -2 z^2 a^{-4} +z^2 a^{-6} -15 z^2-a^3 z-3 a z-3 z a^{-1} -2 z a^{-3} -z a^{-5} +2 a^{-2} + a^{-4} +2
The A2 invariant -q^{12}+q^{10}+q^6+2 q^4-q^2+1- q^{-2} + q^{-6} - q^{-8} + q^{-10} - q^{-12} + q^{-16}
The G2 invariant Data:K11n128/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {9_22,}

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

Vassiliev invariants

V2 and V3: (-1, -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
-4 -8 8 -\frac{14}{3} -\frac{10}{3} 32 \frac{208}{3} \frac{64}{3} 24 -\frac{32}{3} 32 \frac{56}{3} \frac{40}{3} \frac{3809}{30} \frac{502}{15} \frac{4858}{45} -\frac{1025}{18} -\frac{31}{30}

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 K11n128. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
11         11
9        2 -2
7       31 2
5      32  -1
3     43   1
1    44    0
-1   23     -1
-3  24      2
-5 12       -1
-7 2        2
-91         -1
Integral Khovanov Homology

(db, data source)

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