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

K11a193 K11a195

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 12 14 20 16 18 2 22 10 6 8

Alexander Polynomial:

t-4 - 5t-3 + 12t-2 - 18t-1 + 21 - 18t + 12t2 - 5t3 + t4

Conway Polynomial:

1 + z2 + 2z4 + 3z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{K11a106, K11a346, ...}

Determinant and Signature:

{93, -4}

Jones Polynomial:

q-10 - 3q-9 + 6q-8 - 10q-7 + 13q-6 - 15q-5 + 14q-4 - 12q-3 + 10q-2 - 5q-1 + 3 - q

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

{...}

A2 (sl(3)) Invariant:

q-30 + q-24 - 3q-22 + q-20 - 2q-18 - q-16 + q-14 - 3q-12 + 4q-10 + 3q-6 + 2q-4 - q-2 + 1 - q2

HOMFLY-PT Polynomial:

- 4a2z2 - 4a2z4 - a2z6 + 5a4 + 15a4z2 + 14a4z4 + 6a4z6 + a4z8 - 6a6 - 13a6z2 - 9a6z4 - 2a6z6 + 2a8 + 3a8z2 + a8z4

Kauffman Polynomial:

4az3 - 4az5 + az7 - 9a2z2 + 18a2z4 - 13a2z6 + 3a2z8 + a3z + 5a3z5 - 9a3z7 + 3a3z9 + 5a4 - 27a4z2 + 45a4z4 - 33a4z6 + 6a4z8 + a4z10 - 3a5z + 5a5z3 + 7a5z5 - 18a5z7 + 7a5z9 + 6a6 - 25a6z2 + 46a6z4 - 39a6z6 + 10a6z8 + a6z10 - 8a7z + 23a7z3 - 17a7z5 - a7z7 + 4a7z9 + 2a8 - 5a8z2 + 13a8z4 - 14a8z6 + 7a8z8 - 4a9z + 11a9z3 - 12a9z5 + 7a9z7 + a10z2 - 5a10z4 + 5a10z6 - 3a11z3 + 3a11z5 - a12z2 + a12z4

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

{1, 1}

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 11194. 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                   3 1   j = -3                 7 2     j = -5               6 4       j = -7             8 6         j = -9           7 6           j = -11         6 8             j = -13       4 7               j = -15     2 6                 j = -17   1 4                   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, 194]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 194]]
Out[3]=	PD[X[4, 2, 5, 1], X[12, 3, 13, 4], X[14, 6, 15, 5], X[20, 7, 21, 8], > X[16, 9, 17, 10], X[18, 11, 19, 12], X[2, 13, 3, 14], X[22, 16, 1, 15], > X[10, 17, 11, 18], X[6, 19, 7, 20], X[8, 21, 9, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 194]]
Out[4]=	GaussCode[1, -7, 2, -1, 3, -10, 4, -11, 5, -9, 6, -2, 7, -3, 8, -5, 9, -6, 10, > -4, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 194]]
Out[5]=	DTCode[4, 12, 14, 20, 16, 18, 2, 22, 10, 6, 8]
In[6]:=	alex = Alexander[Knot[11, Alternating, 194]][t]
Out[6]=	-4 5 12 18 2 3 4 21 + t - -- + -- - -- - 18 t + 12 t - 5 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 194]][z]
Out[7]=	2 4 6 8 1 + z + 2 z + 3 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 106], Knot[11, Alternating, 194], > Knot[11, Alternating, 346]}
In[9]:=	{KnotDet[Knot[11, Alternating, 194]], KnotSignature[Knot[11, Alternating, 194]]}
Out[9]=	{93, -4}
In[10]:=	J=Jones[Knot[11, Alternating, 194]][q]
Out[10]=	-10 3 6 10 13 15 14 12 10 5 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, 194]}
In[12]:=	A2Invariant[Knot[11, Alternating, 194]][q]
Out[12]=	-30 -24 3 -20 2 -16 -14 3 4 3 2 -2 1 + q + q - --- + q - --- - q + q - --- + --- + -- + -- - q - 22 18 12 10 6 4 q q q q q q 2 > q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 194]][a, z]
Out[13]=	4 6 8 2 2 4 2 6 2 8 2 2 4 5 a - 6 a + 2 a - 4 a z + 15 a z - 13 a z + 3 a z - 4 a z + 4 4 6 4 8 4 2 6 4 6 6 6 4 8 > 14 a z - 9 a z + a z - a z + 6 a z - 2 a z + a z
In[14]:=	Kauffman[Knot[11, Alternating, 194]][a, z]
Out[14]=	4 6 8 3 5 7 9 2 2 4 2 5 a + 6 a + 2 a + a z - 3 a z - 8 a z - 4 a z - 9 a z - 27 a z - 6 2 8 2 10 2 12 2 3 5 3 7 3 > 25 a z - 5 a z + a z - a z + 4 a z + 5 a z + 23 a z + 9 3 11 3 2 4 4 4 6 4 8 4 > 11 a z - 3 a z + 18 a z + 45 a z + 46 a z + 13 a z - 10 4 12 4 5 3 5 5 5 7 5 9 5 > 5 a z + a z - 4 a z + 5 a z + 7 a z - 17 a z - 12 a z + 11 5 2 6 4 6 6 6 8 6 10 6 7 > 3 a z - 13 a z - 33 a z - 39 a z - 14 a z + 5 a z + a z - 3 7 5 7 7 7 9 7 2 8 4 8 6 8 > 9 a z - 18 a z - a z + 7 a z + 3 a z + 6 a z + 10 a z + 8 8 3 9 5 9 7 9 4 10 6 10 > 7 a z + 3 a z + 7 a z + 4 a z + a z + a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 194]], Vassiliev[3][Knot[11, Alternating, 194]]}
Out[15]=	{1, 1}
In[16]:=	Kh[Knot[11, Alternating, 194]][q, t]
Out[16]=	4 7 1 2 1 4 2 6 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 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 7 6 8 7 6 8 6 6 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 3 t t 2 3 3 > --- + -- + 2 q t + q t q q

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a194

K11a193 K11a195