The Alternating Knot 10₂₅

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Acknowledgement

KnotPlot

PD Presentation: X₁₄₂₅ X_5,14,6,15 X_3,13,4,12 X_13,3,14,2 X_11,20,12,1 X_19,6,20,7 X_9,18,10,19 X_7,16,8,17 X_17,8,18,9 X_15,10,16,11

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

DT (Dowker-Thistlethwaite) Code: 4 12 14 16 18 20 2 10 8 6

Minimum Braid Representative:

Length is 11, width is 4
Braid index is 4

A Morse Link Presentation:

3D Invariants:

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

Reversible 2 3 2 / NotAvailable 1

Alexander Polynomial: - 2t^-3 + 8t^-2 - 14t^-1 + 17 - 14t + 8t² - 2t³

Conway Polynomial: 1 - 4z⁴ - 2z⁶

Other knots with the same Alexander/Conway Polynomial: {10₅₆, K11a140, ...}

Determinant and Signature: {65, -4}

Jones Polynomial: q^-10 - 3q^-9 + 6q^-8 - 9q^-7 + 10q^-6 - 11q^-5 + 10q^-4 - 7q^-3 + 5q^-2 - 2q^-1 + 1

Other knots (up to mirrors) with the same Jones Polynomial: {10₅₆, ...}

A2 (sl(3)) Invariant: q^-30 - q^-28 + q^-26 + q^-24 - 2q^-22 + q^-20 - 3q^-18 - q^-12 + 3q^-10 - q^-8 + 2q^-6 + q^-4 + 1

HOMFLY-PT Polynomial: 2a² + 3a²z² + a²z⁴ - 2a⁴z² - 3a⁴z⁴ - a⁴z⁶ - 2a⁶ - 3a⁶z² - 3a⁶z⁴ - a⁶z⁶ + a⁸ + 2a⁸z² + a⁸z⁴

Kauffman Polynomial: - 2a² + 5a²z² - 4a²z⁴ + a²z⁶ + a³z + 4a³z³ - 6a³z⁵ + 2a³z⁷ + 4a⁴z² - 3a⁴z⁴ - 3a⁴z⁶ + 2a⁴z⁸ + 2a⁵z³ - 7a⁵z⁵ + 2a⁵z⁷ + a⁵z⁹ + 2a⁶ - 4a⁶z² + 3a⁶z⁴ - 8a⁶z⁶ + 5a⁶z⁸ - 2a⁷z + 3a⁷z³ - 9a⁷z⁵ + 5a⁷z⁷ + a⁷z⁹ + a⁸ + a⁸z² - 5a⁸z⁴ + a⁸z⁶ + 3a⁸z⁸ + 2a⁹z³ - 5a⁹z⁵ + 5a⁹z⁷ + 3a¹⁰z² - 6a¹⁰z⁴ + 5a¹⁰z⁶ + a¹¹z - 3a¹¹z³ + 3a¹¹z⁵ - a¹²z² + a¹²z⁴

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {0, 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 10₂₅. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

t^rq^j r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2

j = 1 1

j = -1 1

j = -3 4 1

j = -5 4 2

j = -7 6 3

j = -9 5 4

j = -11 5 6

j = -13 4 5

j = -15 2 5

j = -17 1 4

j = -19 2

j = -21 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-28 - 3q^-27 + 2q^-26 + 7q^-25 - 17q^-24 + 7q^-23 + 26q^-22 - 46q^-21 + 11q^-20 + 58q^-19 - 78q^-18 + 8q^-17 + 86q^-16 - 93q^-15 - 4q^-14 + 97q^-13 - 82q^-12 - 18q^-11 + 85q^-10 - 54q^-9 - 24q^-8 + 56q^-7 - 23q^-6 - 19q^-5 + 26q^-4 - 5q^-3 - 9q^-2 + 7q^-1 - 2q + q²

3 q^-54 - 3q^-53 + 2q^-52 + 3q^-51 - q^-50 - 11q^-49 + 5q^-48 + 22q^-47 - 10q^-46 - 41q^-45 + 17q^-44 + 70q^-43 - 21q^-42 - 117q^-41 + 27q^-40 + 174q^-39 - 25q^-38 - 239q^-37 + 9q^-36 + 314q^-35 + 8q^-34 - 372q^-33 - 48q^-32 + 429q^-31 + 79q^-30 - 449q^-29 - 128q^-28 + 459q^-27 + 165q^-26 - 438q^-25 - 203q^-24 + 402q^-23 + 232q^-22 - 351q^-21 - 246q^-20 + 281q^-19 + 261q^-18 - 223q^-17 - 243q^-16 + 144q^-15 + 230q^-14 - 92q^-13 - 186q^-12 + 33q^-11 + 153q^-10 - 7q^-9 - 101q^-8 - 20q^-7 + 71q^-6 + 18q^-5 - 34q^-4 - 21q^-3 + 20q^-2 + 11q^-1 - 6 - 8q + 4q² + 2q³ - 2q⁵ + q⁶

4 q^-88 - 3q^-87 + 2q^-86 + 3q^-85 - 5q^-84 + 5q^-83 - 13q^-82 + 11q^-81 + 16q^-80 - 24q^-79 + 12q^-78 - 42q^-77 + 42q^-76 + 60q^-75 - 77q^-74 + q^-73 - 106q^-72 + 134q^-71 + 180q^-70 - 178q^-69 - 88q^-68 - 262q^-67 + 329q^-66 + 476q^-65 - 286q^-64 - 331q^-63 - 613q^-62 + 593q^-61 + 1015q^-60 - 268q^-59 - 682q^-58 - 1221q^-57 + 758q^-56 + 1697q^-55 - 9q^-54 - 950q^-53 - 1951q^-52 + 677q^-51 + 2241q^-50 + 421q^-49 - 948q^-48 - 2532q^-47 + 374q^-46 + 2427q^-45 + 831q^-44 - 676q^-43 - 2775q^-42 - 16q^-41 + 2234q^-40 + 1105q^-39 - 245q^-38 - 2677q^-37 - 402q^-36 + 1775q^-35 + 1225q^-34 + 233q^-33 - 2296q^-32 - 727q^-31 + 1145q^-30 + 1180q^-29 + 668q^-28 - 1697q^-27 - 898q^-26 + 473q^-25 + 929q^-24 + 919q^-23 - 986q^-22 - 811q^-21 - 57q^-20 + 515q^-19 + 877q^-18 - 369q^-17 - 505q^-16 - 284q^-15 + 123q^-14 + 591q^-13 - 26q^-12 - 178q^-11 - 237q^-10 - 73q^-9 + 272q^-8 + 49q^-7 - 3q^-6 - 102q^-5 - 83q^-4 + 83q^-3 + 20q^-2 + 27q^-1 - 23 - 37q + 20q² + 11q⁴ - 2q⁵ - 10q⁶ + 5q⁷ - q⁸ + 2q⁹ - 2q¹¹ + q¹²

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, 25]]

Out[2]=
PD[X[1, 4, 2, 5], X[5, 14, 6, 15], X[3, 13, 4, 12], X[13, 3, 14, 2], > X[11, 20, 12, 1], X[19, 6, 20, 7], X[9, 18, 10, 19], X[7, 16, 8, 17], > X[17, 8, 18, 9], X[15, 10, 16, 11]]

In[3]:=
GaussCode[Knot[10, 25]]

Out[3]=
GaussCode[-1, 4, -3, 1, -2, 6, -8, 9, -7, 10, -5, 3, -4, 2, -10, 8, -9, 7, -6, > 5]

In[4]:=
DTCode[Knot[10, 25]]

Out[4]=
DTCode[4, 12, 14, 16, 18, 20, 2, 10, 8, 6]

In[5]:=
br = BR[Knot[10, 25]]

Out[5]=
BR[4, {-1, -1, -1, -1, -2, 1, -2, -2, 3, -2, 3}]

In[6]:=
{First[br], Crossings[br]}

Out[6]=
{4, 11}

In[7]:=
BraidIndex[Knot[10, 25]]

Out[7]=
4

In[8]:=
Show[DrawMorseLink[Knot[10, 25]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 25]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

Out[9]=
{Reversible, 2, 3, 2, NotAvailable, 1}

In[10]:=
alex = Alexander[Knot[10, 25]][t]

Out[10]=
2 8 14 2 3 17 - -- + -- - -- - 14 t + 8 t - 2 t 3 2 t t t

In[11]:=
Conway[Knot[10, 25]][z]

Out[11]=
4 6 1 - 4 z - 2 z

In[12]:=
Select[AllKnots[], (alex === Alexander[#][t])&]

Out[12]=
{Knot[10, 25], Knot[10, 56], Knot[11, Alternating, 140]}

In[13]:=
{KnotDet[Knot[10, 25]], KnotSignature[Knot[10, 25]]}

Out[13]=
{65, -4}

In[14]:=
Jones[Knot[10, 25]][q]

Out[14]=
-10 3 6 9 10 11 10 7 5 2 1 + q - -- + -- - -- + -- - -- + -- - -- + -- - - 9 8 7 6 5 4 3 2 q q q q q q q q q

In[15]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]

Out[15]=
{Knot[10, 25], Knot[10, 56]}

In[16]:=
A2Invariant[Knot[10, 25]][q]

Out[16]=
-30 -28 -26 -24 2 -20 3 -12 3 -8 2 -4 1 + q - q + q + q - --- + q - --- - q + --- - q + -- + q 22 18 10 6 q q q q

In[17]:=
HOMFLYPT[Knot[10, 25]][a, z]

Out[17]=
2 6 8 2 2 4 2 6 2 8 2 2 4 4 4 2 a - 2 a + a + 3 a z - 2 a z - 3 a z + 2 a z + a z - 3 a z - 6 4 8 4 4 6 6 6 > 3 a z + a z - a z - a z

In[18]:=
Kauffman[Knot[10, 25]][a, z]

Out[18]=
2 6 8 3 7 11 2 2 4 2 6 2 -2 a + 2 a + a + a z - 2 a z + a z + 5 a z + 4 a z - 4 a z + 8 2 10 2 12 2 3 3 5 3 7 3 9 3 > a z + 3 a z - a z + 4 a z + 2 a z + 3 a z + 2 a z - 11 3 2 4 4 4 6 4 8 4 10 4 12 4 > 3 a z - 4 a z - 3 a z + 3 a z - 5 a z - 6 a z + a z - 3 5 5 5 7 5 9 5 11 5 2 6 4 6 > 6 a z - 7 a z - 9 a z - 5 a z + 3 a z + a z - 3 a z - 6 6 8 6 10 6 3 7 5 7 7 7 9 7 > 8 a z + a z + 5 a z + 2 a z + 2 a z + 5 a z + 5 a z + 4 8 6 8 8 8 5 9 7 9 > 2 a z + 5 a z + 3 a z + a z + a z

In[19]:=
{Vassiliev[2][Knot[10, 25]], Vassiliev[3][Knot[10, 25]]}

Out[19]=
{0, 2}

In[20]:=
Kh[Knot[10, 25]][q, t]

Out[20]=
2 4 1 2 1 4 2 5 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 5 5 6 5 4 6 3 4 t t > ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- + -- + - + 13 4 11 4 11 3 9 3 9 2 7 2 7 5 3 q q t q t q t q t q t q t q t q t q 2 > q t

In[21]:=
ColouredJones[Knot[10, 25], 2][q]

Out[21]=
-28 3 2 7 17 7 26 46 11 58 78 8 86 q - --- + --- + --- - --- + --- + --- - --- + --- + --- - --- + --- + --- - 27 26 25 24 23 22 21 20 19 18 17 16 q q q q q q q q q q q q 93 4 97 82 18 85 54 24 56 23 19 26 5 9 > --- - --- + --- - --- - --- + --- - -- - -- + -- - -- - -- + -- - -- - -- + 15 14 13 12 11 10 9 8 7 6 5 4 3 2 q q q q q q q q q q q q q q 7 2 > - - 2 q + q q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₂₅

10₂₄
10₂₆

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-28 - 3q^-27 + 2q^-26 + 7q^-25 - 17q^-24 + 7q^-23 + 26q^-22 - 46q^-21 + 11q^-20 + 58q^-19 - 78q^-18 + 8q^-17 + 86q^-16 - 93q^-15 - 4q^-14 + 97q^-13 - 82q^-12 - 18q^-11 + 85q^-10 - 54q^-9 - 24q^-8 + 56q^-7 - 23q^-6 - 19q^-5 + 26q^-4 - 5q^-3 - 9q^-2 + 7q^-1 - 2q + q²
3	q^-54 - 3q^-53 + 2q^-52 + 3q^-51 - q^-50 - 11q^-49 + 5q^-48 + 22q^-47 - 10q^-46 - 41q^-45 + 17q^-44 + 70q^-43 - 21q^-42 - 117q^-41 + 27q^-40 + 174q^-39 - 25q^-38 - 239q^-37 + 9q^-36 + 314q^-35 + 8q^-34 - 372q^-33 - 48q^-32 + 429q^-31 + 79q^-30 - 449q^-29 - 128q^-28 + 459q^-27 + 165q^-26 - 438q^-25 - 203q^-24 + 402q^-23 + 232q^-22 - 351q^-21 - 246q^-20 + 281q^-19 + 261q^-18 - 223q^-17 - 243q^-16 + 144q^-15 + 230q^-14 - 92q^-13 - 186q^-12 + 33q^-11 + 153q^-10 - 7q^-9 - 101q^-8 - 20q^-7 + 71q^-6 + 18q^-5 - 34q^-4 - 21q^-3 + 20q^-2 + 11q^-1 - 6 - 8q + 4q² + 2q³ - 2q⁵ + q⁶
4	q^-88 - 3q^-87 + 2q^-86 + 3q^-85 - 5q^-84 + 5q^-83 - 13q^-82 + 11q^-81 + 16q^-80 - 24q^-79 + 12q^-78 - 42q^-77 + 42q^-76 + 60q^-75 - 77q^-74 + q^-73 - 106q^-72 + 134q^-71 + 180q^-70 - 178q^-69 - 88q^-68 - 262q^-67 + 329q^-66 + 476q^-65 - 286q^-64 - 331q^-63 - 613q^-62 + 593q^-61 + 1015q^-60 - 268q^-59 - 682q^-58 - 1221q^-57 + 758q^-56 + 1697q^-55 - 9q^-54 - 950q^-53 - 1951q^-52 + 677q^-51 + 2241q^-50 + 421q^-49 - 948q^-48 - 2532q^-47 + 374q^-46 + 2427q^-45 + 831q^-44 - 676q^-43 - 2775q^-42 - 16q^-41 + 2234q^-40 + 1105q^-39 - 245q^-38 - 2677q^-37 - 402q^-36 + 1775q^-35 + 1225q^-34 + 233q^-33 - 2296q^-32 - 727q^-31 + 1145q^-30 + 1180q^-29 + 668q^-28 - 1697q^-27 - 898q^-26 + 473q^-25 + 929q^-24 + 919q^-23 - 986q^-22 - 811q^-21 - 57q^-20 + 515q^-19 + 877q^-18 - 369q^-17 - 505q^-16 - 284q^-15 + 123q^-14 + 591q^-13 - 26q^-12 - 178q^-11 - 237q^-10 - 73q^-9 + 272q^-8 + 49q^-7 - 3q^-6 - 102q^-5 - 83q^-4 + 83q^-3 + 20q^-2 + 27q^-1 - 23 - 37q + 20q² + 11q⁴ - 2q⁵ - 10q⁶ + 5q⁷ - q⁸ + 2q⁹ - 2q¹¹ + q¹²

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 25]]
Out[2]=	PD[X[1, 4, 2, 5], X[5, 14, 6, 15], X[3, 13, 4, 12], X[13, 3, 14, 2], > X[11, 20, 12, 1], X[19, 6, 20, 7], X[9, 18, 10, 19], X[7, 16, 8, 17], > X[17, 8, 18, 9], X[15, 10, 16, 11]]
In[3]:=	GaussCode[Knot[10, 25]]
Out[3]=	GaussCode[-1, 4, -3, 1, -2, 6, -8, 9, -7, 10, -5, 3, -4, 2, -10, 8, -9, 7, -6, > 5]
In[4]:=	DTCode[Knot[10, 25]]
Out[4]=	DTCode[4, 12, 14, 16, 18, 20, 2, 10, 8, 6]
In[5]:=	br = BR[Knot[10, 25]]
Out[5]=	BR[4, {-1, -1, -1, -1, -2, 1, -2, -2, 3, -2, 3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 25]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 25]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 25]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 3, 2, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 25]][t]
Out[10]=	2 8 14 2 3 17 - -- + -- - -- - 14 t + 8 t - 2 t 3 2 t t t
In[11]:=	Conway[Knot[10, 25]][z]
Out[11]=	4 6 1 - 4 z - 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 25], Knot[10, 56], Knot[11, Alternating, 140]}
In[13]:=	{KnotDet[Knot[10, 25]], KnotSignature[Knot[10, 25]]}
Out[13]=	{65, -4}
In[14]:=	Jones[Knot[10, 25]][q]
Out[14]=	-10 3 6 9 10 11 10 7 5 2 1 + q - -- + -- - -- + -- - -- + -- - -- + -- - - 9 8 7 6 5 4 3 2 q q q q q q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 25], Knot[10, 56]}
In[16]:=	A2Invariant[Knot[10, 25]][q]
Out[16]=	-30 -28 -26 -24 2 -20 3 -12 3 -8 2 -4 1 + q - q + q + q - --- + q - --- - q + --- - q + -- + q 22 18 10 6 q q q q
In[17]:=	HOMFLYPT[Knot[10, 25]][a, z]
Out[17]=	2 6 8 2 2 4 2 6 2 8 2 2 4 4 4 2 a - 2 a + a + 3 a z - 2 a z - 3 a z + 2 a z + a z - 3 a z - 6 4 8 4 4 6 6 6 > 3 a z + a z - a z - a z
In[18]:=	Kauffman[Knot[10, 25]][a, z]
Out[18]=	2 6 8 3 7 11 2 2 4 2 6 2 -2 a + 2 a + a + a z - 2 a z + a z + 5 a z + 4 a z - 4 a z + 8 2 10 2 12 2 3 3 5 3 7 3 9 3 > a z + 3 a z - a z + 4 a z + 2 a z + 3 a z + 2 a z - 11 3 2 4 4 4 6 4 8 4 10 4 12 4 > 3 a z - 4 a z - 3 a z + 3 a z - 5 a z - 6 a z + a z - 3 5 5 5 7 5 9 5 11 5 2 6 4 6 > 6 a z - 7 a z - 9 a z - 5 a z + 3 a z + a z - 3 a z - 6 6 8 6 10 6 3 7 5 7 7 7 9 7 > 8 a z + a z + 5 a z + 2 a z + 2 a z + 5 a z + 5 a z + 4 8 6 8 8 8 5 9 7 9 > 2 a z + 5 a z + 3 a z + a z + a z
In[19]:=	{Vassiliev[2][Knot[10, 25]], Vassiliev[3][Knot[10, 25]]}
Out[19]=	{0, 2}
In[20]:=	Kh[Knot[10, 25]][q, t]
Out[20]=	2 4 1 2 1 4 2 5 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 5 5 6 5 4 6 3 4 t t > ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- + -- + - + 13 4 11 4 11 3 9 3 9 2 7 2 7 5 3 q q t q t q t q t q t q t q t q t q 2 > q t
In[21]:=	ColouredJones[Knot[10, 25], 2][q]
Out[21]=	-28 3 2 7 17 7 26 46 11 58 78 8 86 q - --- + --- + --- - --- + --- + --- - --- + --- + --- - --- + --- + --- - 27 26 25 24 23 22 21 20 19 18 17 16 q q q q q q q q q q q q 93 4 97 82 18 85 54 24 56 23 19 26 5 9 > --- - --- + --- - --- - --- + --- - -- - -- + -- - -- - -- + -- - -- - -- + 15 14 13 12 11 10 9 8 7 6 5 4 3 2 q q q q q q q q q q q q q q 7 2 > - - 2 q + q q

The Alternating Knot 1025

The Alternating Knot 10₂₅