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

Visit K11n103 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X14,5,15,6 X12,8,13,7 X9,20,10,21 X2,11,3,12 X16,13,17,14 X6,15,7,16 X17,22,18,1 X19,8,20,9 X21,18,22,19
Gauss code 1, -6, 2, -1, 3, -8, 4, 10, -5, -2, 6, -4, 7, -3, 8, -7, -9, 11, -10, 5, -11, 9
Dowker-Thistlethwaite code 4 10 14 12 -20 2 16 6 -22 -8 -18
A Braid Representative
A Morse Link Presentation K11n103 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n103/ThurstonBennequinNumber
Hyperbolic Volume 13.4162
A-Polynomial See Data:K11n103/A-polynomial

[edit Notes for K11n103's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n103's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^3+7 t^2-15 t+19-15 t^{-1} +7 t^{-2} - t^{-3}
Conway polynomial -z^6+z^4+4 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 65, -4 }
Jones polynomial 1-3 q^{-1} +6 q^{-2} -8 q^{-3} +11 q^{-4} -11 q^{-5} +10 q^{-6} -8 q^{-7} +5 q^{-8} -2 q^{-9}
HOMFLY-PT polynomial (db, data sources) -2 z^2 a^8-2 a^8+3 z^4 a^6+7 z^2 a^6+3 a^6-z^6 a^4-3 z^4 a^4-3 z^2 a^4-a^4+z^4 a^2+2 z^2 a^2+a^2
Kauffman polynomial (db, data sources) 3 z^3 a^{11}-2 z a^{11}+z^6 a^{10}+4 z^4 a^{10}-3 z^2 a^{10}+3 z^7 a^9-2 z^5 a^9+3 z^3 a^9-z a^9+3 z^8 a^8-2 z^6 a^8+3 z^2 a^8-2 a^8+z^9 a^7+6 z^7 a^7-14 z^5 a^7+7 z^3 a^7-z a^7+6 z^8 a^6-9 z^6 a^6-5 z^4 a^6+9 z^2 a^6-3 a^6+z^9 a^5+6 z^7 a^5-21 z^5 a^5+14 z^3 a^5-3 z a^5+3 z^8 a^4-5 z^6 a^4-4 z^4 a^4+6 z^2 a^4-a^4+3 z^7 a^3-9 z^5 a^3+7 z^3 a^3-z a^3+z^6 a^2-3 z^4 a^2+3 z^2 a^2-a^2
The A2 invariant Data:K11n103/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n103/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n10, K11n144,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11n175,}

Vassiliev invariants

V2 and V3: (4, -9)
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 -72 128 \frac{1112}{3} \frac{160}{3} -1152 -2160 -352 -296 \frac{2048}{3} 2592 \frac{17792}{3} \frac{2560}{3} \frac{193022}{15} \frac{1344}{5} \frac{221888}{45} \frac{1282}{9} \frac{9422}{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=-4 is the signature of K11n103. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
1         11
-1        2 -2
-3       41 3
-5      53  -2
-7     63   3
-9    55    0
-11   56     -1
-13  35      2
-15 25       -3
-17 3        3
-192         -2
Integral Khovanov Homology

(db, data source)

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