From Knot Atlas
Jump to: navigation, search






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

Visit K11n67 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X8493 X14,5,15,6 X2837 X9,19,10,18 X11,17,12,16 X13,20,14,21 X6,15,7,16 X17,11,18,10 X19,1,20,22 X21,12,22,13
Gauss code 1, -4, 2, -1, 3, -8, 4, -2, -5, 9, -6, 11, -7, -3, 8, 6, -9, 5, -10, 7, -11, 10
Dowker-Thistlethwaite code 4 8 14 2 -18 -16 -20 6 -10 -22 -12
A Braid Representative
A Morse Link Presentation K11n67 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number \{1,2\}
3-genus 2
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n67/ThurstonBennequinNumber
Hyperbolic Volume 10.6503
A-Polynomial See Data:K11n67/A-polynomial

[edit Notes for K11n67'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 K11n67's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t+5-2 t^{-1}
Conway polynomial 1-2 z^2
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 9, 0 }
Jones polynomial -q^7+2 q^6-2 q^5+3 q^4-2 q^3+q^2-q+ q^{-1} - q^{-2} + q^{-3}
HOMFLY-PT polynomial (db, data sources) z^4 a^{-4} -z^4+a^2 z^2-z^2 a^{-2} +3 z^2 a^{-4} -z^2 a^{-6} -4 z^2+2 a^2- a^{-2} +3 a^{-4} - a^{-6} -2
Kauffman polynomial (db, data sources) z^9 a^{-3} +z^9 a^{-5} +z^8 a^{-2} +3 z^8 a^{-4} +2 z^8 a^{-6} +a z^7-6 z^7 a^{-3} -4 z^7 a^{-5} +z^7 a^{-7} +a^2 z^6-7 z^6 a^{-2} -18 z^6 a^{-4} -11 z^6 a^{-6} +z^6-5 a z^5-z^5 a^{-1} +10 z^5 a^{-3} +z^5 a^{-5} -5 z^5 a^{-7} -5 a^2 z^4+12 z^4 a^{-2} +31 z^4 a^{-4} +17 z^4 a^{-6} -7 z^4+5 a z^3-z^3 a^{-1} -9 z^3 a^{-3} +3 z^3 a^{-5} +6 z^3 a^{-7} +6 a^2 z^2-8 z^2 a^{-2} -20 z^2 a^{-4} -9 z^2 a^{-6} +9 z^2-a z+2 z a^{-1} +4 z a^{-3} -z a^{-7} -2 a^2+ a^{-2} +3 a^{-4} + a^{-6} -2
The A2 invariant Data:K11n67/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n67/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {6_1, 9_46, K11n97, K11n139,}

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

Vassiliev invariants

V2 and V3: (-2, 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
-8 8 32 \frac{212}{3} \frac{52}{3} -64 -\frac{304}{3} -\frac{160}{3} -24 -\frac{256}{3} 32 -\frac{1696}{3} -\frac{416}{3} -\frac{9751}{15} -\frac{372}{5} -\frac{15844}{45} \frac{343}{9} -\frac{871}{15}

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 K11n67. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
15           1-1
13          1 1
11         11 0
9        21  1
7      111   1
5      12    -1
3    121     0
1   111      -1
-1   12       1
-3 11         0
-5            0
-71           1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1 i=3
r=-4 {\mathbb Z}
r=-3 {\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}
r=-1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}^{2} {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=6 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=7 {\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.