4 1

3_1

5_1

Visit 4 1's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 4 1's page at Knotilus! Visit 4 1's page at the original Knot Atlas! 4_1 is also known as "the Figure Eight knot", as some people think it looks like a figure `8' in one of its common projections. See e.g. [1] . For two 4_1 knots along a closed loop, see 10_59, 10_60, K12a975, and K12a991.

Square depiction	Alternate square depiction	3D depiction	In "figure 8" form
A Neli-Kolam with 3x2 dot array[1]	In curved symmetrical form	Quasi-Celtic depiction	Symmetrical from parametric equation
Thurston's Trick [2]	Cylindrical depiction

Non-prime (compound) versions

• 

  Pseudo-Celtic ornamental knot pattern, with three figure-8 knots along a closed triangular loop.

• 

  (alternative)

• 

  Coat of arms surrounded by figure-8 knots

Knot presentations

Planar diagram presentation	X4251 X8615 X6374 X2738
Gauss code	1, -4, 3, -1, 2, -3, 4, -2
Dowker-Thistlethwaite code	4 6 8 2
Conway Notation	[22]

Minimum Braid Representative: Length is 4, width is 3. Braid index is 3.

A Morse Link Presentation:

Three dimensional invariants

Symmetry type Fully amphicheiral Unknotting number 1 3-genus 1 Bridge index 2 Super bridge index 3 Nakanishi index 1 Maximal Thurston-Bennequin number [-3][-3] Hyperbolic Volume 2.02988 A-Polynomial See Data:4 1/A-polynomial

[edit Notes for 4 1's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus ${\textrm {ConcordanceGenus}}({\textrm {Knot}}(4,1))$ Rasmussen s-Invariant 0

[edit Notes for 4 1's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t-t^{-1}+3$
Conway polynomial	$1-z^{2}$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 5, 0 }
Jones polynomial	$q^{2}+q^{-2}-q-q^{-1}+1$
HOMFLY-PT polynomial (db, data sources)	$a^{2}+a^{-2}-z^{2}-1$
Kauffman polynomial (db, data sources)	$a^{2}z^{2}+z^{2}a^{-2}-a^{2}-a^{-2}+az^{3}+z^{3}a^{-1}-az-za^{-1}+2z^{2}-1$
The A2 invariant	$q^{8}+q^{6}-1+q^{-6}+q^{-8}$
The G2 invariant	$q^{38}+q^{34}-q^{30}+q^{28}+q^{26}+q^{24}+q^{18}+q^{16}-q^{10}-q^{4}-1-q^{-4}-q^{-10}+q^{-16}+q^{-18}+q^{-24}+q^{-26}+q^{-28}-q^{-30}+q^{-34}+q^{-38}$

Further Quantum Invariants

Further quantum knot invariants for 4_1.

A1 Invariants.

Weight	Invariant
1	$q^{5}+q^{-5}$
2	$q^{14}-q^{10}+q^{2}+1+q^{-2}-q^{-10}+q^{-14}$
3	$q^{27}-q^{23}-q^{21}+q^{17}+q^{11}+q^{9}+q^{-9}+q^{-11}+q^{-17}-q^{-21}-q^{-23}+q^{-27}$
4	$q^{44}-q^{40}-q^{38}-q^{36}+q^{34}+q^{32}+q^{30}-q^{26}+q^{24}+q^{22}-q^{18}-q^{16}+q^{4}+q^{2}+1+q^{-2}+q^{-4}-q^{-16}-q^{-18}+q^{-22}+q^{-24}-q^{-26}+q^{-30}+q^{-32}+q^{-34}-q^{-36}-q^{-38}-q^{-40}+q^{-44}$
5	$q^{65}-q^{61}-q^{59}-q^{57}+q^{53}+2q^{51}+q^{49}-q^{45}-q^{43}+q^{39}+q^{37}-q^{35}-2q^{33}-q^{31}+q^{27}+q^{25}+q^{17}+q^{15}+q^{13}+q^{-13}+q^{-15}+q^{-17}+q^{-25}+q^{-27}-q^{-31}-2q^{-33}-q^{-35}+q^{-37}+q^{-39}-q^{-43}-q^{-45}+q^{-49}+2q^{-51}+q^{-53}-q^{-57}-q^{-59}-q^{-61}+q^{-65}$
6	$q^{90}-q^{86}-q^{84}-q^{82}+2q^{76}+2q^{74}+q^{72}-q^{68}-2q^{66}-2q^{64}+q^{62}+q^{60}+q^{58}-q^{54}-2q^{52}-2q^{50}+q^{48}+2q^{46}+2q^{44}+q^{42}-q^{38}-q^{36}+q^{34}+q^{32}+q^{30}-q^{26}-q^{24}-q^{22}+q^{6}+q^{4}+q^{2}+1+q^{-2}+q^{-4}+q^{-6}-q^{-22}-q^{-24}-q^{-26}+q^{-30}+q^{-32}+q^{-34}-q^{-36}-q^{-38}+q^{-42}+2q^{-44}+2q^{-46}+q^{-48}-2q^{-50}-2q^{-52}-q^{-54}+q^{-58}+q^{-60}+q^{-62}-2q^{-64}-2q^{-66}-q^{-68}+q^{-72}+2q^{-74}+2q^{-76}-q^{-82}-q^{-84}-q^{-86}+q^{-90}$

A2 Invariants.

Weight	Invariant
1,0	$q^{8}+q^{6}-1+q^{-6}+q^{-8}$
1,1	$q^{20}+2q^{16}-2q^{10}-2q^{8}+4q^{2}+2+4q^{-2}-2q^{-8}-2q^{-10}+2q^{-16}+q^{-20}$
2,0	$q^{20}+q^{18}+q^{16}-q^{14}-q^{12}-q^{10}-q^{8}+q^{4}+2q^{2}+2+2q^{-2}+q^{-4}-q^{-8}-q^{-10}-q^{-12}-q^{-14}+q^{-16}+q^{-18}+q^{-20}$
3,0	$q^{36}+q^{34}+q^{32}-2q^{28}-2q^{26}-2q^{24}+q^{18}+2q^{16}+3q^{14}+3q^{12}+2q^{10}+q^{8}-q^{4}-2q^{2}-2-2q^{-2}-q^{-4}+q^{-8}+2q^{-10}+3q^{-12}+3q^{-14}+2q^{-16}+q^{-18}-2q^{-24}-2q^{-26}-2q^{-28}+q^{-32}+q^{-34}+q^{-36}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{16}+q^{12}+q^{10}+q^{-10}+q^{-12}+q^{-16}$
1,0,0	$q^{11}+q^{9}+q^{7}-q-q^{-1}+q^{-7}+q^{-9}+q^{-11}$
1,0,1	$q^{26}+2q^{22}+2q^{20}+q^{18}+2q^{16}-2q^{14}-2q^{12}-4q^{10}-4q^{8}+2q^{4}+6q^{2}+7+6q^{-2}+2q^{-4}-4q^{-8}-4q^{-10}-2q^{-12}-2q^{-14}+2q^{-16}+q^{-18}+2q^{-20}+2q^{-22}+q^{-26}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{22}+q^{20}+q^{18}+q^{16}+q^{14}-q^{10}-q^{8}+q^{2}+2+q^{-2}-q^{-8}-q^{-10}+q^{-14}+q^{-16}+q^{-18}+q^{-20}+q^{-22}$
1,0,0,0	$q^{14}+q^{12}+q^{10}+q^{8}-q^{2}-1-q^{-2}+q^{-8}+q^{-10}+q^{-12}+q^{-14}$

B2 Invariants.

Weight	Invariant
0,1	$q^{16}+q^{12}+q^{10}-2+q^{-10}+q^{-12}+q^{-16}$
1,0	$q^{26}+q^{18}+1+q^{-18}+q^{-26}$

B3 Invariants.

Weight	Invariant
1,0,0	$q^{38}+q^{30}+q^{26}+q^{22}-q^{2}+1-q^{-2}+q^{-22}+q^{-26}+q^{-30}+q^{-38}$

B4 Invariants.

Weight	Invariant
1,0,0,0	$q^{50}+q^{42}+q^{38}+q^{34}+q^{30}+q^{26}-q^{6}-q^{2}+1-q^{-2}-q^{-6}+q^{-26}+q^{-30}+q^{-34}+q^{-38}+q^{-42}+q^{-50}$

C3 Invariants.

Weight	Invariant
1,0,0	$q^{22}+q^{18}+q^{16}+q^{14}+q^{12}-q^{2}-2-q^{-2}+q^{-12}+q^{-14}+q^{-16}+q^{-18}+q^{-22}$

C4 Invariants.

Weight	Invariant
1,0,0,0	$q^{28}+q^{24}+q^{22}+q^{20}+q^{18}+q^{16}+q^{14}-q^{4}-q^{2}-2-q^{-2}-q^{-4}+q^{-14}+q^{-16}+q^{-18}+q^{-20}+q^{-22}+q^{-24}+q^{-28}$

D4 Invariants.

Weight	Invariant
0,1,0,0	$q^{38}+q^{34}+3q^{32}+2q^{30}+q^{28}+4q^{26}+q^{24}-2q^{18}-5q^{16}-3q^{14}-3q^{12}-4q^{10}+3q^{6}+5q^{4}+6q^{2}+8+6q^{-2}+5q^{-4}+3q^{-6}-4q^{-10}-3q^{-12}-3q^{-14}-5q^{-16}-2q^{-18}+q^{-24}+4q^{-26}+q^{-28}+2q^{-30}+3q^{-32}+q^{-34}+q^{-38}$
1,0,0,0	$q^{22}+q^{18}+q^{16}+q^{14}+q^{12}-q^{2}-q^{-2}+q^{-12}+q^{-14}+q^{-16}+q^{-18}+q^{-22}$

G2 Invariants.

Weight	Invariant
1,0	$q^{38}+q^{34}-q^{30}+q^{28}+q^{26}+q^{24}+q^{18}+q^{16}-q^{10}-q^{4}-1-q^{-4}-q^{-10}+q^{-16}+q^{-18}+q^{-24}+q^{-26}+q^{-28}-q^{-30}+q^{-34}+q^{-38}$

.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["4 1"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

$-t-t^{-1}+3$

In[5]:=

Conway[K][z]

Out[5]=

$1-z^{2}$

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 5, 0 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

$q^{2}+q^{-2}-q-q^{-1}+1$

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

$a^{2}+a^{-2}-z^{2}-1$

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

$a^{2}z^{2}+z^{2}a^{-2}-a^{2}-a^{-2}+az^{3}+z^{3}a^{-1}-az-za^{-1}+2z^{2}-1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {...}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11n19, ...}

Vassiliev invariants

V2 and V3:

(-1, 0)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 $-4$ $0$ $8$ ${\frac {34}{3}}$ ${\frac {14}{3}}$ $0$ $0$ $0$ $0$ $-{\frac {32}{3}}$ $0$ $-{\frac {136}{3}}$ $-{\frac {56}{3}}$ $-{\frac {1231}{30}}$ ${\frac {142}{15}}$ $-{\frac {1742}{45}}$ ${\frac {79}{18}}$ $-{\frac {271}{30}}$

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). 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 4 1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -2 -1 0 1 2 χ 5         1 1 3           0 1     1 1   0 -1   1 1     0 -3           0 -5 1         1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-2$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=1$ ${\mathbb {Z} }$ $r=2$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the $n+1$-dimensional representation of $sl(2)$)

$n$	$J_{n}$
2	$q^{6}-q^{5}-q^{4}+2q^{3}-q^{2}-q+3-q^{-1}-q^{-2}+2q^{-3}-q^{-4}-q^{-5}+q^{-6}$
3	$q^{12}-q^{11}-q^{10}+2q^{8}-2q^{6}+3q^{4}-3q^{2}+3-3q^{-2}+3q^{-4}-2q^{-6}+2q^{-8}-q^{-10}-q^{-11}+q^{-12}$
4	$q^{20}-q^{19}-q^{18}+3q^{15}-q^{14}-q^{13}-q^{12}-q^{11}+5q^{10}-q^{9}-2q^{8}-2q^{7}-q^{6}+6q^{5}-q^{4}-2q^{3}-2q^{2}-q+7-q^{-1}-2q^{-2}-2q^{-3}-q^{-4}+6q^{-5}-q^{-6}-2q^{-7}-2q^{-8}-q^{-9}+5q^{-10}-q^{-11}-q^{-12}-q^{-13}-q^{-14}+3q^{-15}-q^{-18}-q^{-19}+q^{-20}$
5	$q^{30}-q^{29}-q^{28}+q^{25}+2q^{24}-2q^{22}-q^{21}-q^{20}+q^{19}+3q^{18}+q^{17}-2q^{16}-3q^{15}-2q^{14}+2q^{13}+4q^{12}+2q^{11}-2q^{10}-4q^{9}-2q^{8}+2q^{7}+5q^{6}+2q^{5}-2q^{4}-5q^{3}-2q^{2}+2q+5+2q^{-1}-2q^{-2}-5q^{-3}-2q^{-4}+2q^{-5}+5q^{-6}+2q^{-7}-2q^{-8}-4q^{-9}-2q^{-10}+2q^{-11}+4q^{-12}+2q^{-13}-2q^{-14}-3q^{-15}-2q^{-16}+q^{-17}+3q^{-18}+q^{-19}-q^{-20}-q^{-21}-2q^{-22}+2q^{-24}+q^{-25}-q^{-28}-q^{-29}+q^{-30}$
6	$q^{42}-q^{41}-q^{40}+q^{37}+3q^{35}-q^{34}-2q^{33}-q^{32}-q^{31}+6q^{28}-q^{27}-2q^{26}-2q^{25}-2q^{24}-q^{23}+9q^{21}-2q^{19}-3q^{18}-3q^{17}-2q^{16}+11q^{14}-2q^{12}-4q^{11}-4q^{10}-2q^{9}+12q^{7}-2q^{5}-4q^{4}-4q^{3}-2q^{2}+13-2q^{-2}-4q^{-3}-4q^{-4}-2q^{-5}+12q^{-7}-2q^{-9}-4q^{-10}-4q^{-11}-2q^{-12}+11q^{-14}-2q^{-16}-3q^{-17}-3q^{-18}-2q^{-19}+9q^{-21}-q^{-23}-2q^{-24}-2q^{-25}-2q^{-26}-q^{-27}+6q^{-28}-q^{-31}-q^{-32}-2q^{-33}-q^{-34}+3q^{-35}+q^{-37}-q^{-40}-q^{-41}+q^{-42}$
7	$q^{56}-q^{55}-q^{54}+q^{51}+q^{49}+2q^{48}-q^{47}-2q^{46}-q^{45}-2q^{44}+q^{43}+q^{41}+5q^{40}-2q^{38}-2q^{37}-4q^{36}+2q^{33}+7q^{32}+q^{31}-q^{30}-2q^{29}-7q^{28}-2q^{27}+2q^{25}+9q^{24}+2q^{23}-3q^{21}-9q^{20}-3q^{19}+3q^{17}+10q^{16}+3q^{15}-3q^{13}-10q^{12}-3q^{11}+3q^{9}+11q^{8}+3q^{7}-3q^{5}-11q^{4}-3q^{3}+3q+11+3q^{-1}-3q^{-3}-11q^{-4}-3q^{-5}+3q^{-7}+11q^{-8}+3q^{-9}-3q^{-11}-10q^{-12}-3q^{-13}+3q^{-15}+10q^{-16}+3q^{-17}-3q^{-19}-9q^{-20}-3q^{-21}+2q^{-23}+9q^{-24}+2q^{-25}-2q^{-27}-7q^{-28}-2q^{-29}-q^{-30}+q^{-31}+7q^{-32}+2q^{-33}-4q^{-36}-2q^{-37}-2q^{-38}+5q^{-40}+q^{-41}+q^{-43}-2q^{-44}-q^{-45}-2q^{-46}-q^{-47}+2q^{-48}+q^{-49}+q^{-51}-q^{-54}-q^{-55}+q^{-56}$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 29, 2005, 15:27:48)...
In[2]:=	PD[Knot[4, 1]]
Out[2]=	PD[X[4, 2, 5, 1], X[8, 6, 1, 5], X[6, 3, 7, 4], X[2, 7, 3, 8]]
In[3]:=	GaussCode[Knot[4, 1]]
Out[3]=	GaussCode[1, -4, 3, -1, 2, -3, 4, -2]
In[4]:=	DTCode[Knot[4, 1]]
Out[4]=	DTCode[4, 6, 8, 2]
In[5]:=	br = BR[Knot[4, 1]]
Out[5]=	BR[3, {-1, 2, -1, 2}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{3, 4}
In[7]:=	BraidIndex[Knot[4, 1]]
Out[7]=	3
In[8]:=	Show[DrawMorseLink[Knot[4, 1]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[4, 1]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{FullyAmphicheiral, 1, 1, 2, 3, 1}
In[10]:=	alex = Alexander[Knot[4, 1]][t]
Out[10]=	1 3 - - - t t
In[11]:=	Conway[Knot[4, 1]][z]
Out[11]=	2 1 - z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[4, 1]}
In[13]:=	{KnotDet[Knot[4, 1]], KnotSignature[Knot[4, 1]]}
Out[13]=	{5, 0}
In[14]:=	Jones[Knot[4, 1]][q]
Out[14]=	-2 1 2 1 + q - - - q + q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[4, 1], Knot[11, NonAlternating, 19]}
In[16]:=	A2Invariant[Knot[4, 1]][q]
Out[16]=	-8 -6 6 8 -1 + q + q + q + q
In[17]:=	HOMFLYPT[Knot[4, 1]][a, z]
Out[17]=	-2 2 2 -1 + a + a - z
In[18]:=	Kauffman[Knot[4, 1]][a, z]
Out[18]=	2 3 -2 2 z 2 z 2 2 z 3 -1 - a - a - - - a z + 2 z + -- + a z + -- + a z a 2 a a
In[19]:=	{Vassiliev[2][Knot[4, 1]], Vassiliev[3][Knot[4, 1]]}
Out[19]=	{-1, 0}
In[20]:=	Kh[Knot[4, 1]][q, t]
Out[20]=	1 1 1 5 2 - + q + ----- + --- + q t + q t q 5 2 q t q t
In[21]:=	ColouredJones[Knot[4, 1], 2][q]
Out[21]=	-6 -5 -4 2 -2 1 2 3 4 5 6 3 + q - q - q + -- - q - - - q - q + 2 q - q - q + q 3 q q

See/edit the Rolfsen_Splice_Template.