# 4 1

 (KnotPlot image) See the full Rolfsen Knot Table. Visit 4 1's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 4 1 at Knotilus! 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

### 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 A Morse Link Presentation An Arc Presentation

Length is 4, width is 3,

Braid index is 3

[{3, 5}, {6, 4}, {5, 2}, {1, 3}, {2, 6}, {4, 1}]
 Knot 4_1. A graph, knot 4_1.

### 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

### Four dimensional invariants

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

### Polynomial invariants

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

### "Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, ${\displaystyle q\leftrightarrow q^{-1}}$): {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 ${\displaystyle -4}$ ${\displaystyle 0}$ ${\displaystyle 8}$ ${\displaystyle {\frac {34}{3}}}$ ${\displaystyle {\frac {14}{3}}}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle -{\frac {32}{3}}}$ ${\displaystyle 0}$ ${\displaystyle -{\frac {136}{3}}}$ ${\displaystyle -{\frac {56}{3}}}$ ${\displaystyle -{\frac {1231}{30}}}$ ${\displaystyle {\frac {142}{15}}}$ ${\displaystyle -{\frac {1742}{45}}}$ ${\displaystyle {\frac {79}{18}}}$ ${\displaystyle -{\frac {271}{30}}}$

V2,1 through V6,9 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 ${\displaystyle t^{r}q^{j}}$ are shown, along with their alternating sums ${\displaystyle \chi }$ (fixed ${\displaystyle j}$, alternation over ${\displaystyle r}$). The squares with yellow highlighting are those on the "critical diagonals", where ${\displaystyle j-2r=s+1}$ or ${\displaystyle j-2r=s-1}$, where ${\displaystyle s=}$0 is the signature of 4 1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-2-1012χ
5    11
3     0
1  11 0
-1 11  0
-3     0
-51    1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-1}$ ${\displaystyle i=1}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$