# K11n34

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.

 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

### 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

### 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 \
-6-5-4-3-2-1012345χ
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
 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-3$ $i=-1$ $i=1$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### 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.