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

Visit K11a46 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X8493 X14,5,15,6 X2837 X20,10,21,9 X18,11,19,12 X16,13,17,14 X6,15,7,16 X12,17,13,18 X22,20,1,19 X10,22,11,21
Gauss code 1, -4, 2, -1, 3, -8, 4, -2, 5, -11, 6, -9, 7, -3, 8, -7, 9, -6, 10, -5, 11, -10
Dowker-Thistlethwaite code 4 8 14 2 20 18 16 6 12 22 10
A Braid Representative
A Morse Link Presentation K11a46 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a46/ThurstonBennequinNumber
Hyperbolic Volume 13.0634
A-Polynomial See Data:K11a46/A-polynomial

[edit Notes for K11a46's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant -2

[edit Notes for K11a46's four dimensional invariants]

Polynomial invariants

Alexander polynomial 2 t^3-9 t^2+20 t-25+20 t^{-1} -9 t^{-2} +2 t^{-3}
Conway polynomial 2 z^6+3 z^4+2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 87, 2 }
Jones polynomial q^7-3 q^6+5 q^5-9 q^4+12 q^3-13 q^2+14 q-11+9 q^{-1} -6 q^{-2} +3 q^{-3} - q^{-4}
HOMFLY-PT polynomial (db, data sources) z^6 a^{-2} +z^6-a^2 z^4+3 z^4 a^{-2} -2 z^4 a^{-4} +3 z^4-2 a^2 z^2+5 z^2 a^{-2} -5 z^2 a^{-4} +z^2 a^{-6} +3 z^2-a^2+4 a^{-2} -4 a^{-4} + a^{-6} +1
Kauffman polynomial (db, data sources) z^{10} a^{-2} +z^{10}+3 a z^9+6 z^9 a^{-1} +3 z^9 a^{-3} +3 a^2 z^8+6 z^8 a^{-2} +4 z^8 a^{-4} +5 z^8+a^3 z^7-8 a z^7-14 z^7 a^{-1} -z^7 a^{-3} +4 z^7 a^{-5} -12 a^2 z^6-20 z^6 a^{-2} -z^6 a^{-4} +4 z^6 a^{-6} -27 z^6-4 a^3 z^5-a z^5+z^5 a^{-1} -4 z^5 a^{-3} +z^5 a^{-5} +3 z^5 a^{-7} +14 a^2 z^4+12 z^4 a^{-2} -7 z^4 a^{-4} -3 z^4 a^{-6} +z^4 a^{-8} +29 z^4+5 a^3 z^3+10 a z^3+8 z^3 a^{-1} -2 z^3 a^{-3} -9 z^3 a^{-5} -4 z^3 a^{-7} -6 a^2 z^2+3 z^2 a^{-2} +8 z^2 a^{-4} +z^2 a^{-6} -z^2 a^{-8} -9 z^2-2 a^3 z-4 a z-2 z a^{-1} +4 z a^{-3} +6 z a^{-5} +2 z a^{-7} +a^2-4 a^{-2} -4 a^{-4} - a^{-6} +1
The A2 invariant Data:K11a46/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a46/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_84, K11n184,}

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

Vassiliev invariants

V2 and V3: (2, 0)
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 0 32 \frac{28}{3} -\frac{28}{3} 0 -32 0 0 \frac{256}{3} 0 \frac{224}{3} -\frac{224}{3} -\frac{449}{15} \frac{356}{15} -\frac{3716}{45} \frac{65}{9} -\frac{449}{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=2 is the signature of K11a46. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
15           11
13          2 -2
11         31 2
9        62  -4
7       63   3
5      76    -1
3     76     1
1    58      3
-1   46       -2
-3  25        3
-5 14         -3
-7 2          2
-91           -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
r=-5 {\mathbb Z}
r=-4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=0 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r=1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=6 {\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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