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

Visit K11a255 at Knotilus!

Knot presentations

Planar diagram presentation X6271 X8394 X12,6,13,5 X20,8,21,7 X16,9,17,10 X18,11,19,12 X22,13,1,14 X4,16,5,15 X10,17,11,18 X2,19,3,20 X14,21,15,22
Gauss code 1, -10, 2, -8, 3, -1, 4, -2, 5, -9, 6, -3, 7, -11, 8, -5, 9, -6, 10, -4, 11, -7
Dowker-Thistlethwaite code 6 8 12 20 16 18 22 4 10 2 14
A Braid Representative
A Morse Link Presentation K11a255 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number 1
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a255/ThurstonBennequinNumber
Hyperbolic Volume 17.3074
A-Polynomial See Data:K11a255/A-polynomial

[edit Notes for K11a255's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a255's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^4+6 t^3-17 t^2+30 t-35+30 t^{-1} -17 t^{-2} +6 t^{-3} - t^{-4}
Conway polynomial -z^8-2 z^6-z^4+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 143, -2 }
Jones polynomial q^3-4 q^2+9 q-14+20 q^{-1} -23 q^{-2} +23 q^{-3} -20 q^{-4} +15 q^{-5} -9 q^{-6} +4 q^{-7} - q^{-8}
HOMFLY-PT polynomial (db, data sources) -a^2 z^8+2 a^4 z^6-5 a^2 z^6+z^6-a^6 z^4+7 a^4 z^4-10 a^2 z^4+3 z^4-2 a^6 z^2+8 a^4 z^2-9 a^2 z^2+3 z^2-a^6+3 a^4-3 a^2+2
Kauffman polynomial (db, data sources) 3 a^4 z^{10}+3 a^2 z^{10}+8 a^5 z^9+16 a^3 z^9+8 a z^9+10 a^6 z^8+11 a^4 z^8+9 a^2 z^8+8 z^8+8 a^7 z^7-9 a^5 z^7-40 a^3 z^7-19 a z^7+4 z^7 a^{-1} +4 a^8 z^6-16 a^6 z^6-39 a^4 z^6-42 a^2 z^6+z^6 a^{-2} -22 z^6+a^9 z^5-12 a^7 z^5+a^5 z^5+38 a^3 z^5+15 a z^5-9 z^5 a^{-1} -5 a^8 z^4+10 a^6 z^4+44 a^4 z^4+50 a^2 z^4-2 z^4 a^{-2} +19 z^4-a^9 z^3+5 a^7 z^3+2 a^5 z^3-13 a^3 z^3-6 a z^3+3 z^3 a^{-1} +a^8 z^2-5 a^6 z^2-20 a^4 z^2-22 a^2 z^2-8 z^2-a^7 z-a^5 z+a^3 z+a z+a^6+3 a^4+3 a^2+2
The A2 invariant -q^{24}+q^{22}-2 q^{18}+4 q^{16}-3 q^{14}+2 q^{12}+q^{10}-3 q^8+4 q^6-5 q^4+4 q^2- q^{-2} +3 q^{-4} -2 q^{-6} + q^{-8}
The G2 invariant Data:K11a255/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a79,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a79,}

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
0 -8 0 32 8 0 -\frac{176}{3} \frac{160}{3} -72 0 32 0 0 128 -\frac{824}{3} \frac{664}{3} \frac{304}{3} 32

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 K11a255. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
7           11
5          3 -3
3         61 5
1        83  -5
-1       126   6
-3      129    -3
-5     1111     0
-7    912      3
-9   611       -5
-11  39        6
-13 16         -5
-15 3          3
-171           -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-3 i=-1
r=-7 {\mathbb Z}
r=-6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-4 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-3 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=-2 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=-1 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=0 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{12}
r=1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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