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

Visit K11n3 at Knotilus!

Knot K11n3.
A graph, knot K11n3.
A part of a knot and a part of a graph.

Knot presentations

Planar diagram presentation X4251 X8394 X10,6,11,5 X7,15,8,14 X2,9,3,10 X11,16,12,17 X13,20,14,21 X15,7,16,6 X17,22,18,1 X19,12,20,13 X21,18,22,19
Gauss code 1, -5, 2, -1, 3, 8, -4, -2, 5, -3, -6, 10, -7, 4, -8, 6, -9, 11, -10, 7, -11, 9
Dowker-Thistlethwaite code 4 8 10 -14 2 -16 -20 -6 -22 -12 -18
A Braid Representative
A Morse Link Presentation K11n3 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2\}
3-genus 2
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n3/ThurstonBennequinNumber
Hyperbolic Volume 11.5634
A-Polynomial See Data:K11n3/A-polynomial

[edit Notes for K11n3's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n3's four dimensional invariants]

Polynomial invariants

Alexander polynomial -3 t^2+11 t-15+11 t^{-1} -3 t^{-2}
Conway polynomial -3 z^4-z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 43, -2 }
Jones polynomial q^3-2 q^2+4 q-6+7 q^{-1} -7 q^{-2} +7 q^{-3} -5 q^{-4} +3 q^{-5} - q^{-6}
HOMFLY-PT polynomial (db, data sources) -a^6+3 z^2 a^4+3 a^4-2 z^4 a^2-4 z^2 a^2-2 a^2-z^4-z^2+z^2 a^{-2} + a^{-2}
Kauffman polynomial (db, data sources) a^3 z^9+a z^9+2 a^4 z^8+4 a^2 z^8+2 z^8+a^5 z^7-a^3 z^7+2 z^7 a^{-1} -7 a^4 z^6-13 a^2 z^6+z^6 a^{-2} -5 z^6-a^5 z^5-a^3 z^5-7 a z^5-7 z^5 a^{-1} +3 a^6 z^4+15 a^4 z^4+16 a^2 z^4-4 z^4 a^{-2} +a^7 z^3+4 a^5 z^3+4 a^3 z^3+7 a z^3+6 z^3 a^{-1} -3 a^6 z^2-11 a^4 z^2-10 a^2 z^2+4 z^2 a^{-2} +2 z^2-a^7 z-2 a^5 z-2 a^3 z-3 a z-2 z a^{-1} +a^6+3 a^4+2 a^2- a^{-2}
The A2 invariant Data:K11n3/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n3/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_7, K11a59,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {9_22,}

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{178}{3} \frac{86}{3} 32 -\frac{80}{3} -\frac{32}{3} -8 -\frac{32}{3} 32 -\frac{712}{3} -\frac{344}{3} -\frac{4831}{30} \frac{1462}{15} -\frac{12902}{45} \frac{1087}{18} -\frac{1951}{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=-2 is the signature of K11n3. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
7         11
5        1 -1
3       31 2
1      31  -2
-1     43   1
-3    44    0
-5   33     0
-7  24      2
-9 13       -2
-11 2        2
-131         -1
Integral Khovanov Homology

(db, data source)

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