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

K11a184 K11a186

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 X22,12,1,11 X2,13,3,14 X20,15,21,16 X10,17,11,18 X8,19,9,20 X6,22,7,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 12 14 16 18 22 2 20 10 8 6

Alexander Polynomial:

- 2t-3 + 11t-2 - 25t-1 + 33 - 25t + 11t2 - 2t3

Conway Polynomial:

1 + z2 - z4 - 2z6

Other knots with the same Alexander/Conway Polynomial:

{K11a56, K11a265, ...}

Determinant and Signature:

{109, 0}

Jones Polynomial:

- q-7 + 3q-6 - 6q-5 + 10q-4 - 14q-3 + 17q-2 - 17q-1 + 16 - 12q + 8q2 - 4q3 + q4

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

{K11a265, ...}

A2 (sl(3)) Invariant:

- q-22 + q-18 - 2q-16 + 2q-14 + q-12 - 2q-10 + 3q-8 - 2q-6 + q-4 + q-2 - 1 + 4q2 - 3q4 + q6 + q8 - 2q10 + q12

HOMFLY-PT Polynomial:

a-2z2 + a-2z4 + 1 - z2 - 2z4 - z6 - a2 - 2a2z2 - 2a2z4 - a2z6 + 2a4 + 4a4z2 + 2a4z4 - a6 - a6z2

Kauffman Polynomial:

a-4z4 - 2a-3z3 + 4a-3z5 + 3a-2z2 - 8a-2z4 + 8a-2z6 - a-1z + 5a-1z3 - 12a-1z5 + 10a-1z7 + 1 + 3z2 - 7z4 - 5z6 + 8z8 - 3az + 16az3 - 27az5 + 7az7 + 4az9 + a2 - 7a2z2 + 21a2z4 - 32a2z6 + 12a2z8 + a2z10 - 4a3z + 15a3z3 - 9a3z5 - 11a3z7 + 7a3z9 + 2a4 - 13a4z2 + 34a4z4 - 31a4z6 + 7a4z8 + a4z10 - 4a5z + 11a5z3 - 2a5z5 - 7a5z7 + 3a5z9 + a6 - 6a6z2 + 15a6z4 - 12a6z6 + 3a6z8 - 2a7z + 5a7z3 - 4a7z5 + a7z7

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

{1, -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=0 is the signature of 11185. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 9                       1 j = 7                     3   j = 5                   5 1   j = 3                 7 3     j = 1               9 5       j = -1             9 8         j = -3           8 8           j = -5         6 9             j = -7       4 8               j = -9     2 6                 j = -11   1 4                   j = -13   2                     j = -15 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, 185]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 185]]
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[22, 12, 1, 11], X[2, 13, 3, 14], X[20, 15, 21, 16], > X[10, 17, 11, 18], X[8, 19, 9, 20], X[6, 22, 7, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 185]]
Out[4]=	GaussCode[1, -7, 2, -1, 3, -11, 4, -10, 5, -9, 6, -2, 7, -3, 8, -4, 9, -5, 10, > -8, 11, -6]
In[5]:=	DTCode[Knot[11, Alternating, 185]]
Out[5]=	DTCode[4, 12, 14, 16, 18, 22, 2, 20, 10, 8, 6]
In[6]:=	alex = Alexander[Knot[11, Alternating, 185]][t]
Out[6]=	2 11 25 2 3 33 - -- + -- - -- - 25 t + 11 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 185]][z]
Out[7]=	2 4 6 1 + z - z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 56], Knot[11, Alternating, 185], > Knot[11, Alternating, 265]}
In[9]:=	{KnotDet[Knot[11, Alternating, 185]], KnotSignature[Knot[11, Alternating, 185]]}
Out[9]=	{109, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 185]][q]
Out[10]=	-7 3 6 10 14 17 17 2 3 4 16 - q + -- - -- + -- - -- + -- - -- - 12 q + 8 q - 4 q + q 6 5 4 3 2 q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 185], Knot[11, Alternating, 265]}
In[12]:=	A2Invariant[Knot[11, Alternating, 185]][q]
Out[12]=	-22 -18 2 2 -12 2 3 2 -4 -2 2 4 -1 - q + q - --- + --- + q - --- + -- - -- + q + q + 4 q - 3 q + 16 14 10 8 6 q q q q q 6 8 10 12 > q + q - 2 q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 185]][a, z]
Out[13]=	2 4 2 4 6 2 z 2 2 4 2 6 2 4 z 1 - a + 2 a - a - z + -- - 2 a z + 4 a z - a z - 2 z + -- - 2 2 a a 2 4 4 4 6 2 6 > 2 a z + 2 a z - z - a z
In[14]:=	Kauffman[Knot[11, Alternating, 185]][a, z]
Out[14]=	2 2 4 6 z 3 5 7 2 3 z 1 + a + 2 a + a - - - 3 a z - 4 a z - 4 a z - 2 a z + 3 z + ---- - a 2 a 3 3 2 2 4 2 6 2 2 z 5 z 3 3 3 > 7 a z - 13 a z - 6 a z - ---- + ---- + 16 a z + 15 a z + 3 a a 4 4 5 3 7 3 4 z 8 z 2 4 4 4 6 4 > 11 a z + 5 a z - 7 z + -- - ---- + 21 a z + 34 a z + 15 a z + 4 2 a a 5 5 6 4 z 12 z 5 3 5 5 5 7 5 6 8 z > ---- - ----- - 27 a z - 9 a z - 2 a z - 4 a z - 5 z + ---- - 3 a 2 a a 7 2 6 4 6 6 6 10 z 7 3 7 5 7 > 32 a z - 31 a z - 12 a z + ----- + 7 a z - 11 a z - 7 a z + a 7 7 8 2 8 4 8 6 8 9 3 9 5 9 > a z + 8 z + 12 a z + 7 a z + 3 a z + 4 a z + 7 a z + 3 a z + 2 10 4 10 > a z + a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 185]], Vassiliev[3][Knot[11, Alternating, 185]]}
Out[15]=	{1, -2}
In[16]:=	Kh[Knot[11, Alternating, 185]][q, t]
Out[16]=	8 1 2 1 4 2 6 4 8 - + 9 q + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + q 15 7 13 6 11 6 11 5 9 5 9 4 7 4 7 3 q t q t q t q t q t q t q t q t 6 9 8 8 9 3 3 2 5 2 > ----- + ----- + ----- + ---- + --- + 5 q t + 7 q t + 3 q t + 5 q t + 5 3 5 2 3 2 3 q t q t q t q t q t 5 3 7 3 9 4 > q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a185

K11a184 K11a186