© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:

K11a181 K11a183

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X12,3,13,4 X14,6,15,5 X16,7,17,8 X18,9,19,10 X20,11,21,12 X2,13,3,14 X22,16,1,15 X8,17,9,18 X10,19,11,20 X6,21,7,22

Gauss Code:

{1, -7, 2, -1, 3, -11, 4, -9, 5, -10, 6, -2, 7, -3, 8, -4, 9, -5, 10, -6, 11, -8}

DT (Dowker-Thistlethwaite) Code:

4 12 14 16 18 20 2 22 8 10 6

Alexander Polynomial:

t-4 - 5t-3 + 11t-2 - 13t-1 + 13 - 13t + 11t2 - 5t3 + t4

Conway Polynomial:

1 + 2z2 + z4 + 3z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{73, -4}

Jones Polynomial:

q-10 - 3q-9 + 5q-8 - 8q-7 + 10q-6 - 11q-5 + 11q-4 - 9q-3 + 7q-2 - 4q-1 + 3 - q

Other knots (up to mirrors) with the same Jones Polynomial:

{...}

A2 (sl(3)) Invariant:

q-30 - q-26 - 2q-22 + q-20 - q-18 + q-14 - 2q-12 + 3q-10 + 2q-6 + q-4 + 1 - q2

HOMFLY-PT Polynomial:

- 3a2z2 - 4a2z4 - a2z6 + 4a4 + 13a4z2 + 13a4z4 + 6a4z6 + a4z8 - 4a6 - 11a6z2 - 9a6z4 - 2a6z6 + a8 + 3a8z2 + a8z4

Kauffman Polynomial:

3az3 - 4az5 + az7 - 7a2z2 + 18a2z4 - 14a2z6 + 3a2z8 + a3z - 4a3z3 + 12a3z5 - 12a3z7 + 3a3z9 + 4a4 - 23a4z2 + 42a4z4 - 27a4z6 + 3a4z8 + a4z10 - 11a5z3 + 27a5z5 - 23a5z7 + 6a5z9 + 4a6 - 21a6z2 + 36a6z4 - 24a6z6 + 4a6z8 + a6z10 - 2a7z3 + 5a7z5 - 6a7z7 + 3a7z9 + a8 - 4a8z2 + 7a8z4 - 7a8z6 + 4a8z8 + 2a9z - 2a9z3 - 3a9z5 + 4a9z7 - 4a10z4 + 4a10z6 + a11z - 4a11z3 + 3a11z5 - a12z2 + a12z4

V2 and V3, the type 2 and 3 Vassiliev invariants:

{2, -2}

Khovanov Homology: (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 11182. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 3                       1 j = 1                     2   j = -1                   2 1   j = -3                 5 2     j = -5               5 3       j = -7             6 4         j = -9           5 5           j = -11         5 6             j = -13       3 5               j = -15     2 5                 j = -17   1 3                   j = -19   2                     j = -21 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Show[DrawMorseLink[Knot[11, Alternating, 182]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 182]]
Out[3]=	PD[X[4, 2, 5, 1], X[12, 3, 13, 4], X[14, 6, 15, 5], X[16, 7, 17, 8], > X[18, 9, 19, 10], X[20, 11, 21, 12], X[2, 13, 3, 14], X[22, 16, 1, 15], > X[8, 17, 9, 18], X[10, 19, 11, 20], X[6, 21, 7, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 182]]
Out[4]=	GaussCode[1, -7, 2, -1, 3, -11, 4, -9, 5, -10, 6, -2, 7, -3, 8, -4, 9, -5, 10, > -6, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 182]]
Out[5]=	DTCode[4, 12, 14, 16, 18, 20, 2, 22, 8, 10, 6]
In[6]:=	alex = Alexander[Knot[11, Alternating, 182]][t]
Out[6]=	-4 5 11 13 2 3 4 13 + t - -- + -- - -- - 13 t + 11 t - 5 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 182]][z]
Out[7]=	2 4 6 8 1 + 2 z + z + 3 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 182]}
In[9]:=	{KnotDet[Knot[11, Alternating, 182]], KnotSignature[Knot[11, Alternating, 182]]}
Out[9]=	{73, -4}
In[10]:=	J=Jones[Knot[11, Alternating, 182]][q]
Out[10]=	-10 3 5 8 10 11 11 9 7 4 3 + q - -- + -- - -- + -- - -- + -- - -- + -- - - - q 9 8 7 6 5 4 3 2 q q q q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 182]}
In[12]:=	A2Invariant[Knot[11, Alternating, 182]][q]
Out[12]=	-30 -26 2 -20 -18 -14 2 3 2 -4 2 1 + q - q - --- + q - q + q - --- + --- + -- + q - q 22 12 10 6 q q q q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 182]][a, z]
Out[13]=	4 6 8 2 2 4 2 6 2 8 2 2 4 4 a - 4 a + a - 3 a z + 13 a z - 11 a z + 3 a z - 4 a z + 4 4 6 4 8 4 2 6 4 6 6 6 4 8 > 13 a z - 9 a z + a z - a z + 6 a z - 2 a z + a z
In[14]:=	Kauffman[Knot[11, Alternating, 182]][a, z]
Out[14]=	4 6 8 3 9 11 2 2 4 2 6 2 4 a + 4 a + a + a z + 2 a z + a z - 7 a z - 23 a z - 21 a z - 8 2 12 2 3 3 3 5 3 7 3 9 3 > 4 a z - a z + 3 a z - 4 a z - 11 a z - 2 a z - 2 a z - 11 3 2 4 4 4 6 4 8 4 10 4 12 4 > 4 a z + 18 a z + 42 a z + 36 a z + 7 a z - 4 a z + a z - 5 3 5 5 5 7 5 9 5 11 5 2 6 > 4 a z + 12 a z + 27 a z + 5 a z - 3 a z + 3 a z - 14 a z - 4 6 6 6 8 6 10 6 7 3 7 5 7 > 27 a z - 24 a z - 7 a z + 4 a z + a z - 12 a z - 23 a z - 7 7 9 7 2 8 4 8 6 8 8 8 3 9 > 6 a z + 4 a z + 3 a z + 3 a z + 4 a z + 4 a z + 3 a z + 5 9 7 9 4 10 6 10 > 6 a z + 3 a z + a z + a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 182]], Vassiliev[3][Knot[11, Alternating, 182]]}
Out[15]=	{2, -2}
In[16]:=	Kh[Knot[11, Alternating, 182]][q, t]
Out[16]=	3 5 1 2 1 3 2 5 3 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 5 3 21 8 19 7 17 7 17 6 15 6 15 5 13 5 q q q t q t q t q t q t q t q t 5 5 6 5 5 6 4 5 2 t > ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- + --- + 13 4 11 4 11 3 9 3 9 2 7 2 7 5 3 q t q t q t q t q t q t q t q t q 2 2 t t 2 3 3 > --- + -- + 2 q t + q t q q

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a182

K11a181 K11a183