K11a82

K11a81

K11a83

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a82 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a82/ThurstonBennequinNumber Hyperbolic Volume 13.8883 A-Polynomial See Data:K11a82/A-polynomial

[edit Notes for K11a82's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus [math]\displaystyle{ 4 }[/math] Rasmussen s-Invariant 2

[edit Notes for K11a82's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^4+5 t^3-12 t^2+19 t-21+19 t^{-1} -12 t^{-2} +5 t^{-3} - t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ -z^8-3 z^6-2 z^4+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 95, -2 }
Jones polynomial	[math]\displaystyle{ -q^4+3 q^3-6 q^2+10 q-12+15 q^{-1} -15 q^{-2} +13 q^{-3} -10 q^{-4} +6 q^{-5} -3 q^{-6} + q^{-7} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -a^2 z^8+a^4 z^6-6 a^2 z^6+2 z^6+4 a^4 z^4-14 a^2 z^4-z^4 a^{-2} +9 z^4+5 a^4 z^2-15 a^2 z^2-3 z^2 a^{-2} +13 z^2+2 a^4-6 a^2-2 a^{-2} +7 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a^2 z^{10}+z^{10}+4 a^3 z^9+7 a z^9+3 z^9 a^{-1} +7 a^4 z^8+11 a^2 z^8+3 z^8 a^{-2} +7 z^8+7 a^5 z^7-15 a z^7-7 z^7 a^{-1} +z^7 a^{-3} +5 a^6 z^6-14 a^4 z^6-43 a^2 z^6-12 z^6 a^{-2} -36 z^6+3 a^7 z^5-12 a^5 z^5-21 a^3 z^5-6 a z^5-4 z^5 a^{-1} -4 z^5 a^{-3} +a^8 z^4-5 a^6 z^4+14 a^4 z^4+51 a^2 z^4+15 z^4 a^{-2} +46 z^4-3 a^7 z^3+11 a^5 z^3+28 a^3 z^3+22 a z^3+13 z^3 a^{-1} +5 z^3 a^{-3} -a^8 z^2+a^6 z^2-7 a^4 z^2-27 a^2 z^2-8 z^2 a^{-2} -26 z^2-4 a^5 z-10 a^3 z-10 a z-6 z a^{-1} -2 z a^{-3} +2 a^4+6 a^2+2 a^{-2} +7 }[/math]
The A2 invariant	[math]\displaystyle{ q^{20}-q^{18}+2 q^{16}-q^{14}-q^{12}+q^{10}-4 q^8+2 q^6-2 q^4+2 q^2+3+3 q^{-4} - q^{-6} - q^{-12} }[/math]
The G2 invariant	[math]\displaystyle{ q^{114}-2 q^{112}+4 q^{110}-6 q^{108}+5 q^{106}-3 q^{104}-2 q^{102}+10 q^{100}-17 q^{98}+24 q^{96}-27 q^{94}+19 q^{92}-7 q^{90}-13 q^{88}+37 q^{86}-55 q^{84}+66 q^{82}-62 q^{80}+39 q^{78}-q^{76}-44 q^{74}+91 q^{72}-119 q^{70}+122 q^{68}-90 q^{66}+26 q^{64}+56 q^{62}-124 q^{60}+166 q^{58}-152 q^{56}+86 q^{54}+9 q^{52}-103 q^{50}+148 q^{48}-128 q^{46}+50 q^{44}+54 q^{42}-134 q^{40}+144 q^{38}-84 q^{36}-38 q^{34}+161 q^{32}-239 q^{30}+217 q^{28}-113 q^{26}-48 q^{24}+202 q^{22}-291 q^{20}+282 q^{18}-182 q^{16}+24 q^{14}+130 q^{12}-230 q^{10}+246 q^8-165 q^6+40 q^4+89 q^2-159+157 q^{-2} -70 q^{-4} -45 q^{-6} +148 q^{-8} -183 q^{-10} +139 q^{-12} -24 q^{-14} -108 q^{-16} +209 q^{-18} -228 q^{-20} +170 q^{-22} -58 q^{-24} -69 q^{-26} +156 q^{-28} -184 q^{-30} +152 q^{-32} -79 q^{-34} - q^{-36} +56 q^{-38} -81 q^{-40} +72 q^{-42} -47 q^{-44} +19 q^{-46} +3 q^{-48} -15 q^{-50} +14 q^{-52} -11 q^{-54} +6 q^{-56} -2 q^{-58} + q^{-60} }[/math]

Further Quantum Invariants

K11a82 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["K11a82"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -t^4+5 t^3-12 t^2+19 t-21+19 t^{-1} -12 t^{-2} +5 t^{-3} - t^{-4} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ -z^8-3 z^6-2 z^4+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]=

{ 95, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -q^4+3 q^3-6 q^2+10 q-12+15 q^{-1} -15 q^{-2} +13 q^{-3} -10 q^{-4} +6 q^{-5} -3 q^{-6} + q^{-7} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ -a^2 z^8+a^4 z^6-6 a^2 z^6+2 z^6+4 a^4 z^4-14 a^2 z^4-z^4 a^{-2} +9 z^4+5 a^4 z^2-15 a^2 z^2-3 z^2 a^{-2} +13 z^2+2 a^4-6 a^2-2 a^{-2} +7 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ a^2 z^{10}+z^{10}+4 a^3 z^9+7 a z^9+3 z^9 a^{-1} +7 a^4 z^8+11 a^2 z^8+3 z^8 a^{-2} +7 z^8+7 a^5 z^7-15 a z^7-7 z^7 a^{-1} +z^7 a^{-3} +5 a^6 z^6-14 a^4 z^6-43 a^2 z^6-12 z^6 a^{-2} -36 z^6+3 a^7 z^5-12 a^5 z^5-21 a^3 z^5-6 a z^5-4 z^5 a^{-1} -4 z^5 a^{-3} +a^8 z^4-5 a^6 z^4+14 a^4 z^4+51 a^2 z^4+15 z^4 a^{-2} +46 z^4-3 a^7 z^3+11 a^5 z^3+28 a^3 z^3+22 a z^3+13 z^3 a^{-1} +5 z^3 a^{-3} -a^8 z^2+a^6 z^2-7 a^4 z^2-27 a^2 z^2-8 z^2 a^{-2} -26 z^2-4 a^5 z-10 a^3 z-10 a z-6 z a^{-1} -2 z a^{-3} +2 a^4+6 a^2+2 a^{-2} +7 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_116, K11a7, K11a33,}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {K11a33,}

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["K11a82"];

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+5 t^3-12 t^2+19 t-21+19 t^{-1} -12 t^{-2} +5 t^{-3} - t^{-4} }[/math], [math]\displaystyle{ -q^4+3 q^3-6 q^2+10 q-12+15 q^{-1} -15 q^{-2} +13 q^{-3} -10 q^{-4} +6 q^{-5} -3 q^{-6} + q^{-7} }[/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]=

{10_116, K11a7, K11a33,}

In[6]:=

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

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

Out[6]=

{K11a33,}

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 [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 16 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 16 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -\frac{224}{3} }[/math] [math]\displaystyle{ -\frac{224}{3} }[/math] [math]\displaystyle{ -16 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 352 }[/math] [math]\displaystyle{ \frac{784}{3} }[/math] [math]\displaystyle{ -\frac{208}{3} }[/math] [math]\displaystyle{ \frac{128}{3} }[/math] [math]\displaystyle{ -64 }[/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]-2 is the signature of K11a82. 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 9                       1 -1 7                     2   2 5                   4 1   -3 3                 6 2     4 1               6 4       -2 -1             9 6         3 -3           7 7           0 -5         6 8             -2 -7       4 7               3 -9     2 6                 -4 -11   1 4                   3 -13   2                     -2 -15 1                       1

Integral Khovanov Homology (db, data source)

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

K11a81 K11a83