© | Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table:

10106 10108

KnotPlot

This page is passe. Go here instead!    The Alternating Knot 10107    Visit 10107's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10107's page at Knotilus! Acknowledgement

KnotPlot

PD Presentation:

X1425 X3,12,4,13 X7,14,8,15 X9,19,10,18 X19,7,20,6 X5,17,6,16 X17,11,18,10 X13,8,14,9 X15,1,16,20 X11,2,12,3

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 12 16 14 18 2 8 20 10 6

Minimum Braid Representative: Length is 12, width is 5 Braid index is 5

A Morse Link Presentation:

3D Invariants:

Symmetry Type Unknotting Number 3-Genus Bridge/Super Bridge Index Nakanishi Index Chiral 1 3 3 / NotAvailable 1

Alexander Polynomial:

- t-3 + 8t-2 - 22t-1 + 31 - 22t + 8t2 - t3

Conway Polynomial:

1 + z2 + 2z4 - z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{93, 0}

Jones Polynomial:

- q-5 + 4q-4 - 8q-3 + 12q-2 - 15q-1 + 16 - 14q + 12q2 - 7q3 + 3q4 - q5

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

{...}

A2 (sl(3)) Invariant:

- q-16 + q-14 + 2q-12 - 3q-10 + 2q-8 - q-6 - 2q-4 + 3q-2 - 2 + 4q2 - q4 + q6 + 3q8 - 3q10 + q12 - q16

HOMFLY-PT Polynomial:

- a-4 - a-4z2 + 2a-2 + 3a-2z2 + 2a-2z4 - 2z2 - 2z4 - z6 + 2a2z2 + 2a2z4 - a4z2

Kauffman Polynomial:

a-5z - 2a-5z3 + a-5z5 - a-4 + 3a-4z2 - 5a-4z4 + 3a-4z6 + 3a-3z3 - 7a-3z5 + 5a-3z7 - 2a-2 + 3a-2z2 - 2a-2z4 - 4a-2z6 + 5a-2z8 - 3a-1z + 15a-1z3 - 22a-1z5 + 9a-1z7 + 2a-1z9 + 5z4 - 16z6 + 11z8 - 3az + 17az3 - 27az5 + 11az7 + 2az9 + 2a2z2 - 4a2z4 - 5a2z6 + 6a2z8 - a3z + 6a3z3 - 12a3z5 + 7a3z7 + 2a4z2 - 6a4z4 + 4a4z6 - a5z3 + a5z5

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=0 is the signature of 10107. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 11                     1 j = 9                   2   j = 7                 5 1   j = 5               7 2     j = 3             7 5       j = 1           9 7         j = -1         7 8           j = -3       5 8             j = -5     3 7               j = -7   1 5                 j = -9   3                   j = -11 1

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q-15 - 4q-14 + 3q-13 + 12q-12 - 28q-11 + 5q-10 + 57q-9 - 75q-8 - 18q-7 + 137q-6 - 113q-5 - 70q-4 + 209q-3 - 117q-2 - 121q-1 + 232 - 87q - 141q2 + 194q3 - 38q4 - 120q5 + 115q6 - 68q8 + 41q9 + 8q10 - 21q11 + 8q12 + 2q13 - 3q14 + q15
3	- q-30 + 4q-29 - 3q-28 - 7q-27 + 4q-26 + 23q-25 - 6q-24 - 61q-23 + 6q-22 + 120q-21 + 27q-20 - 215q-19 - 106q-18 + 328q-17 + 250q-16 - 428q-15 - 465q-14 + 484q-13 + 736q-12 - 483q-11 - 1018q-10 + 407q-9 + 1282q-8 - 272q-7 - 1499q-6 + 105q-5 + 1648q-4 + 74q-3 - 1712q-2 - 263q-1 + 1713 + 421q - 1614q2 - 584q3 + 1469q4 + 682q5 - 1225q6 - 764q7 + 966q8 + 756q9 - 665q10 - 706q11 + 414q12 + 573q13 - 196q14 - 426q15 + 60q16 + 280q17 - 152q19 - 24q20 + 76q21 + 17q22 - 32q23 - 9q24 + 14q25 + q26 - 3q27 - 2q28 + 3q29 - q30
4	q-50 - 4q-49 + 3q-48 + 7q-47 - 9q-46 + q-45 - 22q-44 + 26q-43 + 54q-42 - 45q-41 - 34q-40 - 133q-39 + 101q-38 + 305q-37 - 17q-36 - 181q-35 - 661q-34 + 39q-33 + 996q-32 + 597q-31 - 111q-30 - 2035q-29 - 1013q-28 + 1706q-27 + 2416q-26 + 1377q-25 - 3702q-24 - 3850q-23 + 904q-22 + 4758q-21 + 5182q-20 - 3896q-19 - 7599q-18 - 2377q-17 + 5740q-16 + 10202q-15 - 1651q-14 - 10208q-13 - 7006q-12 + 4470q-11 + 14263q-10 + 1973q-9 - 10666q-8 - 10982q-7 + 1818q-6 + 16197q-5 + 5386q-4 - 9408q-3 - 13328q-2 - 1101q-1 + 16069 + 7925q - 7011q2 - 13971q3 - 3925q4 + 14012q5 + 9462q6 - 3593q7 - 12745q8 - 6397q9 + 10039q10 + 9469q11 + 286q12 - 9427q13 - 7525q14 + 4989q15 + 7369q16 + 3065q17 - 4885q18 - 6369q19 + 884q20 + 3898q21 + 3436q22 - 1189q23 - 3636q24 - 782q25 + 1043q26 + 2052q27 + 319q28 - 1281q29 - 609q30 - 90q31 + 694q32 + 330q33 - 260q34 - 151q35 - 149q36 + 138q37 + 96q38 - 45q39 - q40 - 42q41 + 21q42 + 15q43 - 13q44 + 6q45 - 6q46 + 3q47 + 2q48 - 3q49 + q50

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]:=	PD[Knot[10, 107]]
Out[2]=	PD[X[1, 4, 2, 5], X[3, 12, 4, 13], X[7, 14, 8, 15], X[9, 19, 10, 18], > X[19, 7, 20, 6], X[5, 17, 6, 16], X[17, 11, 18, 10], X[13, 8, 14, 9], > X[15, 1, 16, 20], X[11, 2, 12, 3]]
In[3]:=	GaussCode[Knot[10, 107]]
Out[3]=	GaussCode[-1, 10, -2, 1, -6, 5, -3, 8, -4, 7, -10, 2, -8, 3, -9, 6, -7, 4, -5, > 9]
In[4]:=	DTCode[Knot[10, 107]]
Out[4]=	DTCode[4, 12, 16, 14, 18, 2, 8, 20, 10, 6]
In[5]:=	br = BR[Knot[10, 107]]
Out[5]=	BR[5, {-1, -1, 2, -1, 3, 2, 2, -4, 3, -2, 3, -4}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{5, 12}
In[7]:=	BraidIndex[Knot[10, 107]]
Out[7]=	5
In[8]:=	Show[DrawMorseLink[Knot[10, 107]]]
Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 107]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Chiral, 1, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 107]][t]
Out[10]=	-3 8 22 2 3 31 - t + -- - -- - 22 t + 8 t - t 2 t t
In[11]:=	Conway[Knot[10, 107]][z]
Out[11]=	2 4 6 1 + z + 2 z - z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 107]}
In[13]:=	{KnotDet[Knot[10, 107]], KnotSignature[Knot[10, 107]]}
Out[13]=	{93, 0}
In[14]:=	Jones[Knot[10, 107]][q]
Out[14]=	-5 4 8 12 15 2 3 4 5 16 - q + -- - -- + -- - -- - 14 q + 12 q - 7 q + 3 q - q 4 3 2 q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 107]}
In[16]:=	A2Invariant[Knot[10, 107]][q]
Out[16]=	-16 -14 2 3 2 -6 2 3 2 4 6 8 -2 - q + q + --- - --- + -- - q - -- + -- + 4 q - q + q + 3 q - 12 10 8 4 2 q q q q q 10 12 16 > 3 q + q - q
In[17]:=	HOMFLYPT[Knot[10, 107]][a, z]
Out[17]=	2 2 4 -4 2 2 z 3 z 2 2 4 2 4 2 z 2 4 6 -a + -- - 2 z - -- + ---- + 2 a z - a z - 2 z + ---- + 2 a z - z 2 4 2 2 a a a a
In[18]:=	Kauffman[Knot[10, 107]][a, z]
Out[18]=	2 2 3 -4 2 z 3 z 3 3 z 3 z 2 2 4 2 2 z -a - -- + -- - --- - 3 a z - a z + ---- + ---- + 2 a z + 2 a z - ---- + 2 5 a 4 2 5 a a a a a 3 3 4 4 3 z 15 z 3 3 3 5 3 4 5 z 2 z 2 4 > ---- + ----- + 17 a z + 6 a z - a z + 5 z - ---- - ---- - 4 a z - 3 a 4 2 a a a 5 5 5 6 4 4 z 7 z 22 z 5 3 5 5 5 6 3 z > 6 a z + -- - ---- - ----- - 27 a z - 12 a z + a z - 16 z + ---- - 5 3 a 4 a a a 6 7 7 8 4 z 2 6 4 6 5 z 9 z 7 3 7 8 5 z > ---- - 5 a z + 4 a z + ---- + ---- + 11 a z + 7 a z + 11 z + ---- + 2 3 a 2 a a a 9 2 8 2 z 9 > 6 a z + ---- + 2 a z a
In[19]:=	{Vassiliev[2][Knot[10, 107]], Vassiliev[3][Knot[10, 107]]}
Out[19]=	{1, 1}
In[20]:=	Kh[Knot[10, 107]][q, t]
Out[20]=	8 1 3 1 5 3 7 5 8 7 - + 9 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t q t q t 3 3 2 5 2 5 3 7 3 7 4 9 4 > 7 q t + 7 q t + 5 q t + 7 q t + 2 q t + 5 q t + q t + 2 q t + 11 5 > q t
In[21]:=	ColouredJones[Knot[10, 107], 2][q]
Out[21]=	-15 4 3 12 28 5 57 75 18 137 113 70 232 + q - --- + --- + --- - --- + --- + -- - -- - -- + --- - --- - -- + 14 13 12 11 10 9 8 7 6 5 4 q q q q q q q q q q q 209 117 121 2 3 4 5 6 > --- - --- - --- - 87 q - 141 q + 194 q - 38 q - 120 q + 115 q - 3 2 q q q 8 9 10 11 12 13 14 15 > 68 q + 41 q + 8 q - 21 q + 8 q + 2 q - 3 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10107

10106 10108