From Knot Atlas
Jump to: navigation, search






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

Visit K11a189 at Knotilus!

Knot K11a189.
A graph, K11a189.
A part of a link and a part of a graph.

Knot presentations

Planar diagram presentation X4251 X12,3,13,4 X14,6,15,5 X16,7,17,8 X22,10,1,9 X18,11,19,12 X2,13,3,14 X20,16,21,15 X10,17,11,18 X6,19,7,20 X8,22,9,21
Gauss code 1, -7, 2, -1, 3, -10, 4, -11, 5, -9, 6, -2, 7, -3, 8, -4, 9, -6, 10, -8, 11, -5
Dowker-Thistlethwaite code 4 12 14 16 22 18 2 20 10 6 8
A Braid Representative
A Morse Link Presentation K11a189 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:K11a189/ThurstonBennequinNumber
Hyperbolic Volume 17.3742
A-Polynomial See Data:K11a189/A-polynomial

[edit Notes for K11a189'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 K11a189's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-6 t^3+17 t^2-31 t+39-31 t^{-1} +17 t^{-2} -6 t^{-3} + t^{-4}
Conway polynomial z^8+2 z^6+z^4-z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 149, 0 }
Jones polynomial -q^5+4 q^4-9 q^3+16 q^2-21 q+24-24 q^{-1} +21 q^{-2} -15 q^{-3} +9 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^2-8 a^2 z^2-3 z^2 a^{-2} +8 z^2+a^4-2 a^2+2
Kauffman polynomial (db, data sources) 2 a^2 z^{10}+2 z^{10}+6 a^3 z^9+13 a z^9+7 z^9 a^{-1} +7 a^4 z^8+15 a^2 z^8+10 z^8 a^{-2} +18 z^8+4 a^5 z^7-4 a^3 z^7-15 a z^7+z^7 a^{-1} +8 z^7 a^{-3} +a^6 z^6-15 a^4 z^6-42 a^2 z^6-13 z^6 a^{-2} +4 z^6 a^{-4} -43 z^6-9 a^5 z^5-12 a^3 z^5-8 a z^5-17 z^5 a^{-1} -11 z^5 a^{-3} +z^5 a^{-5} -2 a^6 z^4+10 a^4 z^4+34 a^2 z^4+6 z^4 a^{-2} -5 z^4 a^{-4} +33 z^4+6 a^5 z^3+11 a^3 z^3+11 a z^3+13 z^3 a^{-1} +6 z^3 a^{-3} -z^3 a^{-5} +a^6 z^2-3 a^4 z^2-13 a^2 z^2-z^2 a^{-2} +2 z^2 a^{-4} -12 z^2-a^5 z-2 a^3 z-3 a z-3 z a^{-1} -z a^{-3} +a^4+2 a^2+2
The A2 invariant q^{18}-q^{16}+2 q^{12}-4 q^{10}+4 q^8-q^6-q^4+3 q^2-5+5 q^{-2} -3 q^{-4} +2 q^{-6} +3 q^{-8} -3 q^{-10} +2 q^{-12} - q^{-14}
The G2 invariant q^{94}-3 q^{92}+8 q^{90}-16 q^{88}+22 q^{86}-25 q^{84}+14 q^{82}+18 q^{80}-66 q^{78}+128 q^{76}-176 q^{74}+175 q^{72}-102 q^{70}-63 q^{68}+291 q^{66}-498 q^{64}+599 q^{62}-499 q^{60}+178 q^{58}+290 q^{56}-759 q^{54}+1036 q^{52}-973 q^{50}+548 q^{48}+98 q^{46}-728 q^{44}+1080 q^{42}-1000 q^{40}+535 q^{38}+131 q^{36}-693 q^{34}+892 q^{32}-650 q^{30}+51 q^{28}+623 q^{26}-1060 q^{24}+1059 q^{22}-580 q^{20}-201 q^{18}+996 q^{16}-1494 q^{14}+1489 q^{12}-975 q^{10}+111 q^8+783 q^6-1390 q^4+1499 q^2-1079+330 q^{-2} +455 q^{-4} -964 q^{-6} +1000 q^{-8} -594 q^{-10} -53 q^{-12} +636 q^{-14} -876 q^{-16} +682 q^{-18} -137 q^{-20} -501 q^{-22} +962 q^{-24} -1052 q^{-26} +755 q^{-28} -204 q^{-30} -393 q^{-32} +806 q^{-34} -921 q^{-36} +752 q^{-38} -387 q^{-40} -5 q^{-42} +304 q^{-44} -454 q^{-46} +439 q^{-48} -320 q^{-50} +157 q^{-52} -7 q^{-54} -89 q^{-56} +127 q^{-58} -122 q^{-60} +89 q^{-62} -47 q^{-64} +15 q^{-66} +8 q^{-68} -17 q^{-70} +16 q^{-72} -13 q^{-74} +7 q^{-76} -3 q^{-78} + q^{-80}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a30, K11a272,}

Vassiliev invariants

V2 and V3: (-1, 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
-4 8 8 -\frac{14}{3} -\frac{10}{3} -32 -\frac{208}{3} -\frac{160}{3} 8 -\frac{32}{3} 32 \frac{56}{3} \frac{40}{3} \frac{3809}{30} -\frac{1898}{15} \frac{10618}{45} -\frac{449}{18} \frac{1889}{30}

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 K11a189. 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      1310    3
-1     1212     0
-3    912      -3
-5   612       6
-7  39        -6
-9 16         5
-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}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-2 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=-1 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=0 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{13}
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).

Back to the top.