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

Visit K11a300 at Knotilus!

Knot presentations

Planar diagram presentation X6271 X10,4,11,3 X18,5,19,6 X14,7,15,8 X2,10,3,9 X20,12,21,11 X22,13,1,14 X8,15,9,16 X12,18,13,17 X4,19,5,20 X16,21,17,22
Gauss code 1, -5, 2, -10, 3, -1, 4, -8, 5, -2, 6, -9, 7, -4, 8, -11, 9, -3, 10, -6, 11, -7
Dowker-Thistlethwaite code 6 10 18 14 2 20 22 8 12 4 16
A Braid Representative
A Morse Link Presentation K11a300 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number 1
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a300/ThurstonBennequinNumber
Hyperbolic Volume 17.9979
A-Polynomial See Data:K11a300/A-polynomial

[edit Notes for K11a300's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a300's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-6 t^3+17 t^2-32 t+41-32 t^{-1} +17 t^{-2} -6 t^{-3} + t^{-4}
Conway polynomial z^8+2 z^6+z^4-2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 153, 0 }
Jones polynomial -q^5+4 q^4-9 q^3+16 q^2-21 q+25-25 q^{-1} +21 q^{-2} -16 q^{-3} +10 q^{-4} -4 q^{-5} + q^{-6}
HOMFLY-PT polynomial (db, data sources) z^8-2 a^2 z^6-z^6 a^{-2} +5 z^6+a^4 z^4-7 a^2 z^4-3 z^4 a^{-2} +10 z^4+2 a^4 z^2-9 a^2 z^2-3 z^2 a^{-2} +8 z^2+2 a^4-4 a^2+3
Kauffman polynomial (db, data sources) 3 a^2 z^{10}+3 z^{10}+8 a^3 z^9+17 a z^9+9 z^9 a^{-1} +8 a^4 z^8+13 a^2 z^8+11 z^8 a^{-2} +16 z^8+4 a^5 z^7-13 a^3 z^7-35 a z^7-10 z^7 a^{-1} +8 z^7 a^{-3} +a^6 z^6-18 a^4 z^6-49 a^2 z^6-19 z^6 a^{-2} +4 z^6 a^{-4} -53 z^6-8 a^5 z^5+a^3 z^5+20 a z^5-z^5 a^{-1} -11 z^5 a^{-3} +z^5 a^{-5} -2 a^6 z^4+13 a^4 z^4+52 a^2 z^4+15 z^4 a^{-2} -5 z^4 a^{-4} +57 z^4+4 a^5 z^3+2 a^3 z^3+7 z^3 a^{-1} +4 z^3 a^{-3} -z^3 a^{-5} +a^6 z^2-8 a^4 z^2-25 a^2 z^2-6 z^2 a^{-2} +z^2 a^{-4} -23 z^2-a^5 z-2 z a^{-1} -z a^{-3} +2 a^4+4 a^2+3
The A2 invariant q^{18}-q^{16}+q^{14}+3 q^{12}-4 q^{10}+3 q^8-3 q^6-2 q^4+3 q^2-4+6 q^{-2} -3 q^{-4} +2 q^{-6} +3 q^{-8} -3 q^{-10} +2 q^{-12} - q^{-14}
The G2 invariant Data:K11a300/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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

Vassiliev invariants

V2 and V3: (-2, 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
-8 16 32 \frac{68}{3} \frac{28}{3} -128 -\frac{704}{3} -\frac{224}{3} -48 -\frac{256}{3} 128 -\frac{544}{3} -\frac{224}{3} \frac{3089}{15} -\frac{1396}{15} \frac{9476}{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=0 is the signature of K11a300. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
11           1-1
9          3 3
7         61 -5
5        103  7
3       116   -5
1      1410    4
-1     1212     0
-3    913      -4
-5   712       5
-7  39        -6
-9 17         6
-11 3          -3
-131           1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-2 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=-1 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=0 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{14}
r=1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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