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

Visit K11a302 at Knotilus!

Knot presentations

Planar diagram presentation X6271 X10,3,11,4 X18,5,19,6 X14,7,15,8 X20,10,21,9 X4,11,5,12 X22,13,1,14 X8,15,9,16 X12,18,13,17 X2,19,3,20 X16,21,17,22
Gauss code 1, -10, 2, -6, 3, -1, 4, -8, 5, -2, 6, -9, 7, -4, 8, -11, 9, -3, 10, -5, 11, -7
Dowker-Thistlethwaite code 6 10 18 14 20 4 22 8 12 2 16
A Braid Representative
A Morse Link Presentation K11a302 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a302/ThurstonBennequinNumber
Hyperbolic Volume 18.3943
A-Polynomial See Data:K11a302/A-polynomial

[edit Notes for K11a302's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a302's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-7 t^3+20 t^2-33 t+39-33 t^{-1} +20 t^{-2} -7 t^{-3} + t^{-4}
Conway polynomial z^8+z^6-2 z^4+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 161, -4 }
Jones polynomial -q+5-10 q^{-1} +17 q^{-2} -22 q^{-3} +26 q^{-4} -26 q^{-5} +22 q^{-6} -17 q^{-7} +10 q^{-8} -4 q^{-9} + q^{-10}
HOMFLY-PT polynomial (db, data sources) z^4 a^8+2 z^2 a^8+a^8-2 z^6 a^6-6 z^4 a^6-5 z^2 a^6-2 a^6+z^8 a^4+4 z^6 a^4+5 z^4 a^4+2 z^2 a^4-z^6 a^2-2 z^4 a^2+z^2 a^2+2 a^2
Kauffman polynomial (db, data sources) z^4 a^{12}+4 z^5 a^{11}+10 z^6 a^{10}-7 z^4 a^{10}+3 z^2 a^{10}+17 z^7 a^9-24 z^5 a^9+14 z^3 a^9-3 z a^9+19 z^8 a^8-32 z^6 a^8+17 z^4 a^8-5 z^2 a^8+a^8+13 z^9 a^7-12 z^7 a^7-18 z^5 a^7+16 z^3 a^7-5 z a^7+4 z^{10} a^6+20 z^8 a^6-73 z^6 a^6+59 z^4 a^6-18 z^2 a^6+2 a^6+21 z^9 a^5-53 z^7 a^5+29 z^5 a^5-z a^5+4 z^{10} a^4+6 z^8 a^4-46 z^6 a^4+47 z^4 a^4-11 z^2 a^4+8 z^9 a^3-23 z^7 a^3+17 z^5 a^3-z^3 a^3+z a^3+5 z^8 a^2-15 z^6 a^2+13 z^4 a^2-z^2 a^2-2 a^2+z^7 a-2 z^5 a+z^3 a
The A2 invariant q^{30}-q^{28}+3 q^{24}-4 q^{22}+3 q^{20}-3 q^{18}-2 q^{16}+3 q^{14}-5 q^{12}+6 q^{10}-3 q^8+2 q^6+3 q^4-2 q^2+3- q^{-2}
The G2 invariant Data:K11a302/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: (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 -48 16 0 \frac{64}{3} -\frac{224}{3} -48 0 128 0 0 456 -\frac{272}{3} \frac{1760}{3} \frac{200}{3} 88

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=-4 is the signature of K11a302. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
3           1-1
1          4 4
-1         61 -5
-3        114  7
-5       127   -5
-7      1410    4
-9     1212     0
-11    1014      -4
-13   712       5
-15  310        -7
-17 17         6
-19 3          -3
-211           1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-5 i=-3
r=-8 {\mathbb Z}
r=-7 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-5 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-4 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=-3 {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=-2 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{14} {\mathbb Z}^{14}
r=-1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=0 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{11}
r=1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=3 {\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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