From Knot Atlas
Jump to: navigation, search






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

Visit K11n34 at Knotilus!

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

K11n34 is not k-colourable for any k. See The Determinant and the Signature.

Gateknot.jpg Knot emblem on the closed gate of the mathematics department at night. Cambridge, England. See also Heegaard Floer Knot Homology.
Knot K11n34.
A graph, knot K11n34.
A part of a knot and a part of a graph.

Knot presentations

Planar diagram presentation X4251 X8493 X12,5,13,6 X2837 X9,17,10,16 X11,18,12,19 X6,13,7,14 X15,20,16,21 X17,1,18,22 X19,14,20,15 X21,10,22,11
Gauss code 1, -4, 2, -1, 3, -7, 4, -2, -5, 11, -6, -3, 7, 10, -8, 5, -9, 6, -10, 8, -11, 9
Dowker-Thistlethwaite code 4 8 12 2 -16 -18 6 -20 -22 -14 -10
A Braid Representative
A Morse Link Presentation K11n34 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number 1
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n34/ThurstonBennequinNumber
Hyperbolic Volume 11.2191
A-Polynomial See Data:K11n34/A-polynomial

[edit Notes for K11n34's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus [0,3]
Rasmussen s-Invariant 0

[edit Notes for K11n34's four dimensional invariants] By the theorem of M. Freedman, the topological 4-genus is zero, as the Alexander polynomial is one.

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:K11n34/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {0_1, K11n42,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11n42,}

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 K11n34. 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).

Back to the top.