K11a112

K11a111

K11a113

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a112 at Knotilus!

Knot presentations

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

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number [math]\displaystyle{ \{1,2\} }[/math] 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a112/ThurstonBennequinNumber Hyperbolic Volume 16.343 A-Polynomial See Data:K11a112/A-polynomial

[edit Notes for K11a112's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus [math]\displaystyle{ 1 }[/math] Rasmussen s-Invariant 0

[edit Notes for K11a112's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^4-6 t^3+15 t^2-25 t+31-25 t^{-1} +15 t^{-2} -6 t^{-3} + t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ z^8+2 z^6-z^4-3 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 125, 0 }
Jones polynomial	[math]\displaystyle{ -q^5+4 q^4-8 q^3+14 q^2-18 q+20-20 q^{-1} +17 q^{-2} -12 q^{-3} +7 q^{-4} -3 q^{-5} + q^{-6} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^8-2 a^2 z^6-z^6 a^{-2} +5 z^6+a^4 z^4-8 a^2 z^4-3 z^4 a^{-2} +9 z^4+3 a^4 z^2-10 a^2 z^2-2 z^2 a^{-2} +6 z^2+2 a^4-3 a^2+ a^{-2} +1 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ 2 a^2 z^{10}+2 z^{10}+5 a^3 z^9+11 a z^9+6 z^9 a^{-1} +5 a^4 z^8+7 a^2 z^8+8 z^8 a^{-2} +10 z^8+3 a^5 z^7-10 a^3 z^7-24 a z^7-4 z^7 a^{-1} +7 z^7 a^{-3} +a^6 z^6-13 a^4 z^6-30 a^2 z^6-10 z^6 a^{-2} +4 z^6 a^{-4} -30 z^6-8 a^5 z^5+6 a^3 z^5+19 a z^5-6 z^5 a^{-1} -10 z^5 a^{-3} +z^5 a^{-5} -3 a^6 z^4+12 a^4 z^4+39 a^2 z^4-6 z^4 a^{-4} +30 z^4+5 a^5 z^3-2 a^3 z^3-3 a z^3+8 z^3 a^{-1} +3 z^3 a^{-3} -z^3 a^{-5} +2 a^6 z^2-8 a^4 z^2-21 a^2 z^2+3 z^2 a^{-2} +2 z^2 a^{-4} -10 z^2-a^5 z-a z-3 z a^{-1} -z a^{-3} +2 a^4+3 a^2- a^{-2} +1 }[/math]
The A2 invariant	[math]\displaystyle{ q^{18}+2 q^{12}-3 q^{10}+3 q^8-q^6-2 q^4+2 q^2-5+4 q^{-2} -2 q^{-4} +2 q^{-6} +3 q^{-8} -2 q^{-10} +2 q^{-12} - q^{-14} }[/math]
The G2 invariant	[math]\displaystyle{ q^{94}-2 q^{92}+5 q^{90}-9 q^{88}+11 q^{86}-12 q^{84}+5 q^{82}+11 q^{80}-32 q^{78}+59 q^{76}-80 q^{74}+82 q^{72}-56 q^{70}-11 q^{68}+115 q^{66}-222 q^{64}+297 q^{62}-283 q^{60}+156 q^{58}+70 q^{56}-331 q^{54}+533 q^{52}-570 q^{50}+403 q^{48}-73 q^{46}-309 q^{44}+581 q^{42}-627 q^{40}+434 q^{38}-68 q^{36}-299 q^{34}+502 q^{32}-461 q^{30}+179 q^{28}+200 q^{26}-506 q^{24}+596 q^{22}-418 q^{20}+36 q^{18}+410 q^{16}-746 q^{14}+838 q^{12}-646 q^{10}+216 q^8+289 q^6-705 q^4+874 q^2-737+363 q^{-2} +107 q^{-4} -487 q^{-6} +626 q^{-8} -488 q^{-10} +146 q^{-12} +231 q^{-14} -463 q^{-16} +455 q^{-18} -206 q^{-20} -151 q^{-22} +463 q^{-24} -584 q^{-26} +485 q^{-28} -208 q^{-30} -138 q^{-32} +410 q^{-34} -527 q^{-36} +476 q^{-38} -288 q^{-40} +60 q^{-42} +137 q^{-44} -258 q^{-46} +281 q^{-48} -233 q^{-50} +141 q^{-52} -39 q^{-54} -40 q^{-56} +83 q^{-58} -94 q^{-60} +78 q^{-62} -47 q^{-64} +20 q^{-66} +3 q^{-68} -14 q^{-70} +15 q^{-72} -13 q^{-74} +7 q^{-76} -3 q^{-78} + q^{-80} }[/math]

Further Quantum Invariants

K11a112 Quantum Invariants.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["K11a112"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

[math]\displaystyle{ t^4-6 t^3+15 t^2-25 t+31-25 t^{-1} +15 t^{-2} -6 t^{-3} + t^{-4} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ z^8+2 z^6-z^4-3 z^2+1 }[/math]

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

[math]\displaystyle{ \{1\} }[/math]

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 125, 0 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

[math]\displaystyle{ -q^5+4 q^4-8 q^3+14 q^2-18 q+20-20 q^{-1} +17 q^{-2} -12 q^{-3} +7 q^{-4} -3 q^{-5} + q^{-6} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

[math]\displaystyle{ z^8-2 a^2 z^6-z^6 a^{-2} +5 z^6+a^4 z^4-8 a^2 z^4-3 z^4 a^{-2} +9 z^4+3 a^4 z^2-10 a^2 z^2-2 z^2 a^{-2} +6 z^2+2 a^4-3 a^2+ a^{-2} +1 }[/math]

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

[math]\displaystyle{ 2 a^2 z^{10}+2 z^{10}+5 a^3 z^9+11 a z^9+6 z^9 a^{-1} +5 a^4 z^8+7 a^2 z^8+8 z^8 a^{-2} +10 z^8+3 a^5 z^7-10 a^3 z^7-24 a z^7-4 z^7 a^{-1} +7 z^7 a^{-3} +a^6 z^6-13 a^4 z^6-30 a^2 z^6-10 z^6 a^{-2} +4 z^6 a^{-4} -30 z^6-8 a^5 z^5+6 a^3 z^5+19 a z^5-6 z^5 a^{-1} -10 z^5 a^{-3} +z^5 a^{-5} -3 a^6 z^4+12 a^4 z^4+39 a^2 z^4-6 z^4 a^{-4} +30 z^4+5 a^5 z^3-2 a^3 z^3-3 a z^3+8 z^3 a^{-1} +3 z^3 a^{-3} -z^3 a^{-5} +2 a^6 z^2-8 a^4 z^2-21 a^2 z^2+3 z^2 a^{-2} +2 z^2 a^{-4} -10 z^2-a^5 z-a z-3 z a^{-1} -z a^{-3} +2 a^4+3 a^2- a^{-2} +1 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {K11a5,}

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["K11a112"];

In[4]:=

{A = Alexander[K][t], J = Jones[K][q]}

KnotTheory::loading: Loading precomputed data in PD4Knots`.

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[4]=

{ [math]\displaystyle{ t^4-6 t^3+15 t^2-25 t+31-25 t^{-1} +15 t^{-2} -6 t^{-3} + t^{-4} }[/math], [math]\displaystyle{ -q^5+4 q^4-8 q^3+14 q^2-18 q+20-20 q^{-1} +17 q^{-2} -12 q^{-3} +7 q^{-4} -3 q^{-5} + q^{-6} }[/math] }

In[5]:=

DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]=

{}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{K11a5,}

Vassiliev invariants

V2 and V3:

(-3, 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 [math]\displaystyle{ -12 }[/math] [math]\displaystyle{ 16 }[/math] [math]\displaystyle{ 72 }[/math] [math]\displaystyle{ 98 }[/math] [math]\displaystyle{ 46 }[/math] [math]\displaystyle{ -192 }[/math] [math]\displaystyle{ -\frac{1088}{3} }[/math] [math]\displaystyle{ -\frac{320}{3} }[/math] [math]\displaystyle{ -80 }[/math] [math]\displaystyle{ -288 }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ -1176 }[/math] [math]\displaystyle{ -552 }[/math] [math]\displaystyle{ -\frac{6991}{10} }[/math] [math]\displaystyle{ \frac{862}{5} }[/math] [math]\displaystyle{ -\frac{3994}{5} }[/math] [math]\displaystyle{ \frac{293}{2} }[/math] [math]\displaystyle{ -\frac{2031}{10} }[/math]

V_2,1 through V_6,9 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 [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s-1 }[/math], where [math]\displaystyle{ s= }[/math]0 is the signature of K11a112. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 11                       1 -1 9                     3   3 7                   5 1   -4 5                 9 3     6 3               9 5       -4 1             11 9         2 -1           10 10           0 -3         7 10             -3 -5       5 10               5 -7     2 7                 -5 -9   1 5                   4 -11   2                     -2 -13 1                       1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ i=1 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]

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.

K11a111 K11a113