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

Visit K11n123 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X5,16,6,17 X7,15,8,14 X12,10,13,9 X2,11,3,12 X13,18,14,19 X15,21,16,20 X17,22,18,1 X19,8,20,9 X21,7,22,6
Gauss code 1, -6, 2, -1, -3, 11, -4, 10, 5, -2, 6, -5, -7, 4, -8, 3, -9, 7, -10, 8, -11, 9
Dowker-Thistlethwaite code 4 10 -16 -14 12 2 -18 -20 -22 -8 -6
A Braid Representative
A Morse Link Presentation K11n123 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 2
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n123/ThurstonBennequinNumber
Hyperbolic Volume 13.524
A-Polynomial See Data:K11n123/A-polynomial

[edit Notes for K11n123's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 2
Rasmussen s-Invariant 0

[edit Notes for K11n123's four dimensional invariants]

Polynomial invariants

Alexander polynomial 3 t^2-14 t+23-14 t^{-1} +3 t^{-2}
Conway polynomial 3 z^4-2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 57, 0 }
Jones polynomial -q^3+4 q^2-6 q+9-10 q^{-1} +9 q^{-2} -8 q^{-3} +6 q^{-4} -3 q^{-5} + q^{-6}
HOMFLY-PT polynomial (db, data sources) a^6-3 z^2 a^4-2 a^4+2 z^4 a^2+3 z^2 a^2+2 a^2+z^4-z^2-1-z^2 a^{-2} + a^{-2}
Kauffman polynomial (db, data sources) a^3 z^9+a z^9+3 a^4 z^8+5 a^2 z^8+2 z^8+3 a^5 z^7+5 a^3 z^7+3 a z^7+z^7 a^{-1} +a^6 z^6-5 a^4 z^6-8 a^2 z^6-2 z^6-9 a^5 z^5-20 a^3 z^5-9 a z^5+2 z^5 a^{-1} -3 a^6 z^4-6 a^4 z^4-4 a^2 z^4+4 z^4 a^{-2} +3 z^4+7 a^5 z^3+14 a^3 z^3+6 a z^3+z^3 a^{-3} +3 a^6 z^2+8 a^4 z^2+7 a^2 z^2-2 z^2 a^{-2} -2 a^5 z-2 a^3 z-a^6-2 a^4-2 a^2- a^{-2} -1
The A2 invariant Data:K11n123/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n123/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: (-2, 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
-8 16 32 -\frac{28}{3} -\frac{20}{3} -128 -\frac{320}{3} -\frac{128}{3} 16 -\frac{256}{3} 128 \frac{224}{3} \frac{160}{3} \frac{5729}{15} -\frac{436}{15} \frac{12596}{45} -\frac{449}{9} \frac{929}{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 K11n123. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
7         1-1
5        3 3
3       31 -2
1      63  3
-1     54   -1
-3    45    -1
-5   45     1
-7  24      -2
-9 14       3
-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}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=0 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{6}
r=1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=3 {\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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