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

Visit K11a219 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X12,3,13,4 X18,5,19,6 X16,7,17,8 X20,9,21,10 X14,12,15,11 X2,13,3,14 X22,15,1,16 X6,17,7,18 X10,19,11,20 X8,21,9,22
Gauss code 1, -7, 2, -1, 3, -9, 4, -11, 5, -10, 6, -2, 7, -6, 8, -4, 9, -3, 10, -5, 11, -8
Dowker-Thistlethwaite code 4 12 18 16 20 14 2 22 6 10 8
A Braid Representative
A Morse Link Presentation K11a219 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2\}
3-genus 2
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a219/ThurstonBennequinNumber
Hyperbolic Volume 13.4578
A-Polynomial See Data:K11a219/A-polynomial

[edit Notes for K11a219's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a219's four dimensional invariants]

Polynomial invariants

Alexander polynomial -4 t^2+22 t-35+22 t^{-1} -4 t^{-2}
Conway polynomial -4 z^4+6 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 87, -2 }
Jones polynomial q-3+6 q^{-1} -10 q^{-2} +13 q^{-3} -13 q^{-4} +14 q^{-5} -11 q^{-6} +8 q^{-7} -5 q^{-8} +2 q^{-9} - q^{-10}
HOMFLY-PT polynomial (db, data sources) -a^{10}+2 z^2 a^8-z^4 a^6+2 z^2 a^6+a^6-2 z^4 a^4+a^4-z^4 a^2+z^2 a^2+z^2
Kauffman polynomial (db, data sources) z^7 a^{11}-5 z^5 a^{11}+8 z^3 a^{11}-4 z a^{11}+2 z^8 a^{10}-8 z^6 a^{10}+9 z^4 a^{10}-3 z^2 a^{10}+a^{10}+2 z^9 a^9-5 z^7 a^9+3 z^3 a^9+z^{10} a^8+2 z^8 a^8-14 z^6 a^8+14 z^4 a^8-5 z^2 a^8+6 z^9 a^7-18 z^7 a^7+20 z^5 a^7-17 z^3 a^7+6 z a^7+z^{10} a^6+7 z^8 a^6-27 z^6 a^6+30 z^4 a^6-13 z^2 a^6-a^6+4 z^9 a^5-5 z^7 a^5+2 z a^5+7 z^8 a^4-16 z^6 a^4+19 z^4 a^4-9 z^2 a^4+a^4+7 z^7 a^3-12 z^5 a^3+9 z^3 a^3+5 z^6 a^2-5 z^4 a^2+z^2 a^2+3 z^5 a-3 z^3 a+z^4-z^2
The A2 invariant Data:K11a219/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a219/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: (6, -15)
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
24 -120 288 812 172 -2880 -5776 -960 -1176 2304 7200 19488 4128 \frac{205431}{5} -\frac{15924}{5} \frac{110988}{5} 499 \frac{17271}{5}

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 K11a219. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
3           11
1          2 -2
-1         41 3
-3        73  -4
-5       63   3
-7      77    0
-9     76     1
-11    47      3
-13   47       -3
-15  14        3
-17 14         -3
-19 1          1
-211           -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-3 i=-1
r=-9 {\mathbb Z}
r=-8 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-7 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-5 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-4 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=0 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=2 {\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

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See/edit the Hoste-Thistlethwaite Knot Page master template (intermediate).

See/edit the Hoste-Thistlethwaite_Splice_Base (expert).

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