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

Visit K11n55 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X8493 X5,14,6,15 X2837 X16,10,17,9 X18,11,19,12 X13,6,14,7 X22,16,1,15 X20,17,21,18 X12,19,13,20 X10,22,11,21
Gauss code 1, -4, 2, -1, -3, 7, 4, -2, 5, -11, 6, -10, -7, 3, 8, -5, 9, -6, 10, -9, 11, -8
Dowker-Thistlethwaite code 4 8 -14 2 16 18 -6 22 20 12 10
A Braid Representative
A Morse Link Presentation K11n55 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n55/ThurstonBennequinNumber
Hyperbolic Volume 12.8462
A-Polynomial See Data:K11n55/A-polynomial

[edit Notes for K11n55'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 K11n55's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^3+6 t^2-14 t+19-14 t^{-1} +6 t^{-2} - t^{-3}
Conway polynomial -z^6+z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 61, 0 }
Jones polynomial q^6-3 q^5+5 q^4-8 q^3+10 q^2-10 q+10-7 q^{-1} +5 q^{-2} -2 q^{-3}
HOMFLY-PT polynomial (db, data sources) -z^6 a^{-2} -4 z^4 a^{-2} +z^4 a^{-4} +3 z^4-2 a^2 z^2-7 z^2 a^{-2} +2 z^2 a^{-4} +8 z^2-2 a^2-4 a^{-2} + a^{-4} +6
Kauffman polynomial (db, data sources) z^9 a^{-1} +z^9 a^{-3} +6 z^8 a^{-2} +3 z^8 a^{-4} +3 z^8+3 a z^7+5 z^7 a^{-1} +5 z^7 a^{-3} +3 z^7 a^{-5} +a^2 z^6-14 z^6 a^{-2} -7 z^6 a^{-4} +z^6 a^{-6} -5 z^6-4 a z^5-15 z^5 a^{-1} -21 z^5 a^{-3} -10 z^5 a^{-5} +4 a^2 z^4+12 z^4 a^{-2} +2 z^4 a^{-4} -3 z^4 a^{-6} +11 z^4+3 a^3 z^3+8 a z^3+15 z^3 a^{-1} +19 z^3 a^{-3} +9 z^3 a^{-5} -5 a^2 z^2-10 z^2 a^{-2} +2 z^2 a^{-6} -13 z^2-2 a^3 z-5 a z-7 z a^{-1} -6 z a^{-3} -2 z a^{-5} +2 a^2+4 a^{-2} + a^{-4} +6
The A2 invariant Data:K11n55/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n55/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {9_33,}

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{80}{3} \frac{64}{3} -8 \frac{32}{3} 32 \frac{56}{3} \frac{40}{3} \frac{2911}{30} \frac{1178}{15} -\frac{538}{45} \frac{161}{18} -\frac{449}{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 K11n55. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
13         11
11        2 -2
9       31 2
7      52  -3
5     53   2
3    55    0
1   55     0
-1  36      3
-3 24       -2
-5 3        3
-72         -2
Integral Khovanov Homology

(db, data source)

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