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

Visit K11n42 at Knotilus!

K11n42 is the mirror of the "Kinoshita-Terasaka" knot; it is a mutant of the (mirror of the) Conway knot K11n34. See also Heegaard Floer Knot Homology.

K11n42 is not k-colourable for any k. See The Determinant and the Signature.

Knot K11n42.
A graph, knot K11n42.
A part of a knot and a part of a graph.

Knot presentations

Planar diagram presentation X4251 X8493 X12,5,13,6 X2837 X9,18,10,19 X11,21,12,20 X6,13,7,14 X15,10,16,11 X17,22,18,1 X19,15,20,14 X21,16,22,17
Gauss code 1, -4, 2, -1, 3, -7, 4, -2, -5, 8, -6, -3, 7, 10, -8, 11, -9, 5, -10, 6, -11, 9
Dowker-Thistlethwaite code 4 8 12 2 -18 -20 6 -10 -22 -14 -16
A Braid Representative
A Morse Link Presentation K11n42 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number 1
3-genus 2
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n42/ThurstonBennequinNumber
Hyperbolic Volume 11.2191
A-Polynomial See Data:K11n42/A-polynomial

[edit Notes for K11n42's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n42's four dimensional invariants]

Polynomial invariants

Alexander polynomial 1
Conway polynomial 1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 1, 0 }
Jones polynomial -q^4+2 q^3-2 q^2+2 q+ q^{-2} -2 q^{-3} +2 q^{-4} -2 q^{-5} + q^{-6}
HOMFLY-PT polynomial (db, data sources) -a^2 z^6+z^6+a^4 z^4-6 a^2 z^4-z^4 a^{-2} +6 z^4+3 a^4 z^2-11 a^2 z^2-3 z^2 a^{-2} +11 z^2+2 a^4-6 a^2-2 a^{-2} +7
Kauffman polynomial (db, data sources) a z^9+z^9 a^{-1} +a^4 z^8+2 a^2 z^8+2 z^8 a^{-2} +3 z^8+2 a^5 z^7+2 a^3 z^7-5 a z^7-4 z^7 a^{-1} +z^7 a^{-3} +a^6 z^6-4 a^4 z^6-14 a^2 z^6-11 z^6 a^{-2} -20 z^6-9 a^5 z^5-12 a^3 z^5-2 z^5 a^{-1} -5 z^5 a^{-3} -4 a^6 z^4+2 a^4 z^4+26 a^2 z^4+16 z^4 a^{-2} +36 z^4+9 a^5 z^3+16 a^3 z^3+12 a z^3+11 z^3 a^{-1} +6 z^3 a^{-3} +3 a^6 z^2-2 a^4 z^2-20 a^2 z^2-9 z^2 a^{-2} -24 z^2-3 a^5 z-7 a^3 z-7 a z-5 z a^{-1} -2 z a^{-3} +2 a^4+6 a^2+2 a^{-2} +7
The A2 invariant q^{18}+q^{14}-q^{12}-q^{10}-q^8-2 q^6+q^4+3+2 q^{-2} + q^{-4} + q^{-6} - q^{-8} - q^{-12}
The G2 invariant Data:K11n42/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {0_1, K11n34,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11n34,}

Vassiliev invariants

V2 and V3: (0, 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
0 16 0 -16 0 0 -\frac{128}{3} -\frac{128}{3} -16 0 128 0 0 312 \frac{32}{3} 176 8 8

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 K11n42. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
9           1-1
7          1 1
5         11 0
3       121  0
1      211   2
-1     132    0
-3    221     1
-5   111      -1
-7  121       0
-9 11         0
-11 1          -1
-131           1
Integral Khovanov Homology

(db, data source)

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