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

Visit K11a290 at Knotilus!

Knot presentations

Planar diagram presentation X6271 X10,3,11,4 X16,5,17,6 X20,8,21,7 X18,9,19,10 X2,11,3,12 X8,13,9,14 X22,16,1,15 X4,17,5,18 X12,19,13,20 X14,22,15,21
Gauss code 1, -6, 2, -9, 3, -1, 4, -7, 5, -2, 6, -10, 7, -11, 8, -3, 9, -5, 10, -4, 11, -8
Dowker-Thistlethwaite code 6 10 16 20 18 2 8 22 4 12 14
A Braid Representative
A Morse Link Presentation K11a290 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number 1
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a290/ThurstonBennequinNumber
Hyperbolic Volume 17.3442
A-Polynomial See Data:K11a290/A-polynomial

[edit Notes for K11a290'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 K11a290's four dimensional invariants]

Polynomial invariants

Alexander polynomial -3 t^3+15 t^2-32 t+41-32 t^{-1} +15 t^{-2} -3 t^{-3}
Conway polynomial -3 z^6-3 z^4+z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 141, 0 }
Jones polynomial q^4-4 q^3+9 q^2-15 q+20-22 q^{-1} +23 q^{-2} -19 q^{-3} +14 q^{-4} -9 q^{-5} +4 q^{-6} - q^{-7}
HOMFLY-PT polynomial (db, data sources) -z^2 a^6-a^6+3 z^4 a^4+6 z^2 a^4+2 a^4-2 z^6 a^2-6 z^4 a^2-6 z^2 a^2-a^2-z^6-z^4+z^2+1+z^4 a^{-2} +z^2 a^{-2}
Kauffman polynomial (db, data sources) 3 a^4 z^{10}+3 a^2 z^{10}+6 a^5 z^9+16 a^3 z^9+10 a z^9+4 a^6 z^8+5 a^4 z^8+16 a^2 z^8+15 z^8+a^7 z^7-17 a^5 z^7-41 a^3 z^7-9 a z^7+14 z^7 a^{-1} -13 a^6 z^6-39 a^4 z^6-60 a^2 z^6+9 z^6 a^{-2} -25 z^6-3 a^7 z^5+10 a^5 z^5+22 a^3 z^5-14 a z^5-19 z^5 a^{-1} +4 z^5 a^{-3} +13 a^6 z^4+45 a^4 z^4+54 a^2 z^4-7 z^4 a^{-2} +z^4 a^{-4} +14 z^4+3 a^7 z^3+3 a^5 z^3+2 a^3 z^3+12 a z^3+9 z^3 a^{-1} -z^3 a^{-3} -5 a^6 z^2-17 a^4 z^2-19 a^2 z^2+2 z^2 a^{-2} -5 z^2-a^7 z-a^5 z-a^3 z-2 a z-z a^{-1} +a^6+2 a^4+a^2+1
The A2 invariant Data:K11a290/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a290/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a100,}

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

Vassiliev invariants

V2 and V3: (1, -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
4 -16 8 \frac{206}{3} \frac{82}{3} -64 -\frac{640}{3} \frac{32}{3} -112 \frac{32}{3} 128 \frac{824}{3} \frac{328}{3} \frac{22591}{30} -\frac{4382}{15} \frac{26462}{45} \frac{2177}{18} \frac{3391}{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=0 is the signature of K11a290. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
9           11
7          3 -3
5         61 5
3        93  -6
1       116   5
-1      1210    -2
-3     1110     1
-5    812      4
-7   611       -5
-9  38        5
-11 16         -5
-13 3          3
-151           -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 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}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-3 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-2 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=-1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=0 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{11}
r=1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
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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